physics is easy and fun

Destroy The wall 2

2012-01-08T04:26:00.000-08:00

Use a very limited amount of balls at which to throw at the cubes, in hopes of knocking them all down.

http://www.flashphysicsgames.com

Cargo Bridge

2009-06-22T17:39:00.000-07:00

cargobridge is a fun game that, in our in our ability to issue a charge on the rigid object, harmony, and have a simple mathematical calculation. in this game we basically have to move certain objects from the object, but for all that across we need to first create a strong bridge, so that the things that can mean in the move.

Download

A Classic Propellor Toy

2009-02-20T16:29:00.000-08:00

This classic toy was well known before Leonardo da Vinci was a boy, and may have influenced some of his aerodynamic ideas. There are also stories about Orville and Wilbur Wright playing with this toy as kids.

The toy is easy to make, being nothing more than a propellor on a stick, but the physics behind its stability in flight are not so simple.

To make the toy, we need the following:

A block of soft pine, 8 inches long, 2 inches wide, and 1/2 inch thick. The dimensions are not critical.
A 10 inch dowel, 1/4 inch in diameter.
A drill or auger with a 1/4 inch bit.
A wood file or shaping tool, or a whittling knife. Power tools like a drum sander or belt sander make the job go much faster.
A drop of white glue.

Click on image for a larger picture

We start by drilling a 1/4 inch wide hole through the 8 inch block of soft pine.

Click on image for a larger picture

Next, we remove the wood from the corners of the block.

If you are using a knife, hold the block in your left hand, and shave away the wood on the right side of the block. To make the propellor shape, we are removing only the wood on the top right side. The left side is untouched, and the right side is shaved down to a sharp edge.

Click on image for a larger picture

Now turn the block over, and repeat, shaving off the right side only, so the propellor blade is a thin piece of wood, at a pronounced angle to the hole.

Click on image for a larger picture

Now hold the wood block by the blade you have just made, and carve the other end of the propellor in just the same way as the first. Again, only the right side is shaved down to the bottom, and the left side is unshaved.

A knife, while traditional, is not the fastest, easiest, or safest way to remove the wood. Using a wood file or a shaving tool or planer is better. Power sanders are even faster.

The wood can be left in its rough whittled form, or it can be sanded smooth. You can paint the blades, or draw designs on them with felt tipped markers.

Click on image for a larger picture

Now we glue the dowel into the hole. In the photo, I am using a dowel that is 9½ inches long. The dowel can be a little shorter or a little longer, but a shorter dowel will make a less stable flight, and a longer dowel adds unneeded weight. The optimum length is something you will want to experiment with, since each hand carved toy will be slightly different.

How to fly it

Hold the dowel against your left palm using your right fingertips.

Click on image for a larger picture

Now quickly slide your right hand forward and your left hand back, so your left fingertips are against your right palm. The propellor toy will fly away, and land a short walk away.

Click on image for a larger picture

The photo above shows some toys made by hand in Africa. We support the artisans there by offering these hand painted toys in our catalog.

Destroy The Wall

2009-02-20T16:17:00.000-08:00

Knock all of the cubes off of the various platforms by using a variety of approaches like tossing straight shot or bouncing it off of the walls. Remember, you lose control of the ball's movement as soon as it crosses the red line.

copyright 2008 Flash Physics Games | Contact: blognog [at] hotmail.com

Swinging Ball (theory)

2009-01-18T03:03:00.000-08:00

Consider a light string of length l that connects a small ball of mass M and a unmovable nail. The ball is hang by the string. Then the ball is given an initial speed of v, so that it starts swinging. The gravity constant is g.We want the ball pass through the nail. Determine the minimum speed of v in order to doing this! (It's not necessary to get the string in tension at all time).

WORK, ENERGY & POWER

2009-01-09T20:25:00.000-08:00

WORK, ENERGY & POWER

Work is a very normal term, usually one we dislike; "clean up your room !", "mow the lawn !" etc.

This word of "work" brings to mind pushing, pulling, walking back and forth. The very unusual thing about the day-to-day usage is that it is almost identical to the Physics usage of the term.

"The WORK done on an object, is the product of the average force on it and the distance travelled in the direction of the force."

Notice; the work is done on an object, like a lump of wood during wood stacking, by something which exerts a force ( you on the wood ). This force must then proceed to move it through a distance in its direction.

You are stacking wood.

In section A, lifting the wood, you are doing work on the log as the force you exert is in the same direction as the distance travelled.

In section B, apart from a slight amount of force to start moving it along the dotted line, you are doing very little work on the log as the lifting force you exert is not in the direction of travel.

In section C, gravity does work on the log.

In VERY simplistic terms

The unit of work in the modern system is the joule J . ( Very old units include the calorie, BTU and the erg. )

GRAPHICALLY

Work has no sense of direction. We do not ascribe arrows to work or energy.

Distance is used rather than displacement in the simple definition because the force acting may take a windy path. You are literally doing work on the pen when you push it writing. The total path taken which is important is the distance rather than the displacement.

"ENERGY is the ABILITY of an object to do work for whatever reason."

This again sounds like common sense, but stored energy in whatever form has the same units as work and can do, numerically, that amount of work.

Energy comes in various forms;

chemical eg nitroglycerine, or food - indeed the amount of energy involved in exothermic reactions is measured in joules as is nutritional energy values of foods.

heat - both the heat associated with water and the radiation heat associated with the warm sunlight.

motion - a ball thrown hard onto your flesh certainly exerts a force into your skin through a distance. This particular energy is easy to measure and is called kinetic energy.

"hidden" energies called potential energies. A spring in a set mouse trap has one such energy, as has an old tree limb waiting to fall down on someone's head.

Interchangeability of the energies ;

Like momentum, the work-energy idea turns out to be a conservation law. Whenever a process occurs, energy does work and turns into a new form of energy or energies.

When all the forms of energy before and after any process are added we find exactly the same number.

PRINCIPLE OF CONSERVATION OF ENERGY; " In any closed system, the total amount of energy remains constant regardless of any process which takes place."

Again, physicists would like to know why, - it is linked to momentum and mass is also a form of energy. ( OK - what is energy ? )

GRAVITATIONAL POTENTIAL ENERGY;

In falling through a height "h" which is in the same direction as the force, the work done by gravity is

work done = force.dist = Mg.h

thus Grav. Pot. Energy Ep = Mgh

This is a stored energy available to be converted into movement energy on release. The Hydro uses this energy in the form of stored water which is released, converts first to kinetic energy then to electrical energy which is distributed around the State.

KINETIC ENERGY; " Energy available because of the object's motion".

Consider a mass, m, which is moving with a speed , v, and does work which brings it to rest.

The unbalanced force, F, which it exerts in doing the work is, by Newton's Third Law also exerted on it , bringing it to a halt.

F_unbal = ma

so, Work done = F_unbal. dist = mas

( we are assuming all of this takes place in a straight line so that distance and displacement are essentially the same )

Using 2as = v²- v_o²

we get mas = 1/2 .mv² = Work done

E_k = Kinetic energy = 1/2 mv²

All forms of energy can have such formulae worked out for them !

Eg 1; A swing oscillates through a height of 3m. How fast is the little girl going at the bottom of the swing ?

Soln; This movement is not in a straight line so we must rely on conservation of energy to see how fast the girl is going. We must assume that no energy is turned into heat or other less easily calculated forms.

In swinging, the energy changes from Grav. Pot . Energy to Kinetic Energy. So

E_p lost = E_k gained ( cons of energy )

mgh lost = 1/2 .mv² gained

thus, gh = 1/2 v² ( as the mass is common )

9.8 . 3 = 0.5 v²

v² = 58.7

v = 7.65 ms^-1

Eg 2; A 4 kg stone is thrown from the top of a hill which is 20m high. It is thrown at 30 ms^-1 at angle such that its maximum height reached is 15m.

a) How fast is it travelling at the top ?

b) How fast is travelling when it reaches the bottom ?

Soln; a) We could do this problem by the conventional projectile motion but because it only involves energy changes, that is a far simpler method.

The total energy of the stone at the start of the journey is composed of Kinetic Energy if we start by ignoring that it is above the sea.

Total Energy at start = E_k

= 1/2 mv²

= 0.5 . 4 . 900 = 1800 J

When it rises, 15m it loses kinetic energy but gains pot energy.

E_p gained = mgh' = 4 . 9.8 . 15 = 588 J

At the top we have a mixture of energies = starting energy

thus 1800 = 588 + new E_k

new E_k = 1800 - 588 = 1212 J = 1/2 m(v_new)²

thus v_new = 24.6 ms^-1 = speed at the top.

b) At the bottom of the cliff, it has lost additional E_p which is converted into E_k

Additional E_k = mgh" = 4 . 9.8 . 20 = 784 J

new total energy is now = 1800 + 784 = 2584 J

This is now all kinetic energy, so 2584 = 1/2 m(v_bottom)²

v_bottom = 36 ms^-1 near enough.

In every such operation, however, we usually lose some energy in undesirable forms, usually heat generated by friction or some such process.

Heat is not easily turned back into "useful" forms of energy. All of a car's petrol energy eventually turns into heat; much of in the first place out of the exhaust system, some into warming the surrounding air through drag, some in warming the oil in the various parts through friction and lastly in the brakes through friction.

Energy efficient buses try to avoid the latter loss by using some form of energy storage device for example a gyro ( storing kinetic energy ) or electrical generators for converting the vehicle's kinetic energy back to electrical energy.

Ironically, in our houses, we generate heat in stoves, hot water tanks and heaters from the Gravitational Pot. Energy of the stored water. Better insulation lessens the loss of such energy to the outside. Many other techniques exist for decreasing a home's reliance on Hydro energy. We pay, of course, for the Hydro energy we use. The electrical companies use a variant of the joule called a kilowatt-hour.

Most of our food energy is used to generate heat. This provides the conditions for our body cells to flourish in. Spare energy from this is available for doing our day-to-day activities and any left over goes into stored chemical energy called fat.

POWER

"Power is the rate of doing work or changing energy."

P = Work Done = ΔEnergy

t t

A powerful person is capable of doing the same work as a less powerful person in a shorter time.

The unit of power is the watt, W which is the Js^-1.

Eg ; If Poatina generates 500 MW at 90% efficiency from a head of water 1000m above the generator, how much water is needed each second ?

Soln; The water clearly loses Grav Pot Energy so that this is the energy change we need.

P = Work done = Δenergy
t t

500 x 106 = mgh = m . 9.8 . 1000
t t

thus, m / t = 5.1 x 10⁴ kg s^-1 = 51 tonnes s^-1

But the station is only 90% efficient, so the required amount of water is

= 51 x 100 / 90 = 56.7 tonnes s^-1

Return to Phys tutes list

PROBLEMS

1. You push a table through 3m with a force of 30N. How much work have you done on the table? The table fails to accelerate continuously due to friction. What form of energy is created? ( 90 J)

2. In lifting a 20kg bucket of water through 2m from a hole, work has been done and energy transformed. What work have you done, where have you obtained the energy from and what form of energy has the water and bucket now got? (392J)

3. You are writing an English essay of total length 3 pages. Estimate how far the pen moves in your script and how much force you apply to the pen on average. Hence estimate how much work you do on the pen. Where does the physical (not mental! ) energy go that you expend? (Is there such a quantity as mental energy?) ( ~ 4J)

4. A major environmental push is for "energy conservation" in the house, work place etc. How does this conception differ from the pure physicists' conception of energy conservation?

5. Comets, in their highly elliptical orbits, travel fastest near the Sun and slowest out beyond Jupiter. Discuss the energy changes in such an orbit.

Building a radio in 10 minutes.

2009-01-09T20:09:00.000-08:00

For our 10 minute radio, we will need these parts:

A ferrite loop antenna coil
In our other crystal radios we wound the coil by hand. In this project we use a much smaller coil with a ferrite rod inside, from our catalog. The ferrite rod allows the coil to be smaller, and it can be moved in and out of the coil for coarse tuning.
A variable capacitor (30 to 160 picofarads)
We carry this in our catalog. You can also find them in old broken or discarded radios.
A Germanium diode (1N34A)
We carry this in our catalog.
A piezoelectric earphone
Also in our catalog.
Two alligator jumper wires
We use alligator jumper wires here for convenience. They are used to connect the ground and antenna wires to a good ground and a long wire antenna. We carry these in our catalog.
About 50 to 100 feet of stranded insulated wire for an antenna.
This is actually optional, since you can use a TV antenna or FM radio antenna by connecting our radio to one of the lead-in wires. But it's fun to throw your own wire up over a tree or on top of a house, and it makes the radio a little more portable.
A block of wood or something similar for a base

Click on photo for a larger picture

You can see from the photo how simple this radio is, and why it can be put together in a very short time.

The black painted wire from the ferrite loop is soldered to the center lead of the variable capacitor. The unpainted wire is soldered to the rightmost lead of the variable capacitor.

The germanium diode is soldered to the rightmost lead of the variable capacitor.

One of the piezoelectric earphone wires is soldered to the free end of the germanium diode. The other is soldered to the center lead of the variable capacitor.

The red painted wire of the coil is attached to the long wire antenna with an alligator clip lead.

The green painted wire of the coil is attached to a good ground (such as a cold water pipe) using another alligator clip lead.

That's it -- you're done!

Click on photo for a larger picture

How does it work?

We will start the tuning with the variable capacitor set in the middle of its range, neither all the way clockwise, nor all the way counter clockwise.

With the earphone in your ear, slowly move the ferrite rod into the coil, listening for radio stations.

With a long antenna, you can tune several radio stations. In some areas, one or two stations will be so close or so powerful that they overwhelm all the others, and you will only hear those one or two stations.

How does the ferrite change the frequency?

The ferrite rod increases the inductance of the coil. In our other (hand-wound) coils, we increased the inductance by winding some more loops, or by using a "tapped" coil, and selecting a tap that was farther down the coil.

As the ferrite rod is inserted into the coil, more of the coil is affected by the ferrite, and so the inductance increases. Increasing the inductance moves the frequency lower. This allows us to hear stations "lower on the radio dial".

Ferrite is used because it is magnetic, like iron or steel, but it is not a conductor of electricity. If it were conductive, the coil would induce "eddy currents" in it, and some of the energy would be lost heating up the core. Because ferrite is not a conductor, we can use its magnetic properties to change the inductance of the coil, without losing volume.

If you have a long antenna, a good ground, and you are not too close to a strong station, the variable capacitor will help in fine tuning the stations.

There are actually two coils of wire wound around the ferrite rod. The large coil is connected to the variable capacitor. The small coil is connected to the antenna and ground.

This arrangement allows the radio to be more selective, so that strong stations don't drown out the weak ones. Really strong local stations will still overwhelm the more distant stations, however.

If you have no strong local stations, you can make the stations you receive sound louder by connecting the antenna and ground directly to the large coil. Connect the antenna to the center lead of the variable capacitor, and the ground to the rightmost lead of the variable capacitor. The stations will be louder, but they will most likely all be heard at once, since you radio will be less selective in tuning out adjacent stations.

Wave Motion

2009-01-09T20:07:00.000-08:00

There are three main ways in which wave motion differs from the motion of objects made of matter.

Superposition

The first, and most profound, difference between wave motion and the motion of objects is that waves do not display any repulsion of each other analogous to the normal forces between objects that come in contact. Two wave patterns can therefore overlap in the same region of space, as shown in the figure at the top of the page. Where the two waves coincide, they add together. For instance, suppose that at a certain location in at a certain moment in time, each wave would have had a crest 3 cm above the normal water level. The waves combine at this point to make a 6-cm crest. We use negative numbers to represent depressions in the water. If both waves would have had a troughs measuring -3 cm, then they combine to make an extradeep -6 cm trough. A +3 cm crest and a -3 cm trough result in a height of zero, i.e. the waves momentarily cancel each other out at that point. This additive rule is referred to as the principle of superposition, "superposition" being merely a fancy word for "adding."

Superposition can occur not just with sinusoidal waves like the ones in the figure above but with waves of any shape. The figures on the following page show superposition of wave pulses. A pulse is simply a wave of very short duration. These pulses consist only of a single hump or trough. If you hit a clothesline sharply, you will observe pulses heading off in both directions. This is analogous to the way ripples spread out in all directions when you make a disturbance at one point on water. The same occurs when the hammer on a piano comes up and hits a string.

Experiments to date have not shown any deviation from the principle of superposition in the case of light waves. For other types of waves, it is typically a very good approximation for low-energy waves.

Discussion Questions

In figure (c) below, the fifth frame shows the spring just about perfectly flat. If the two pulses have essentially canceled each other out perfectly, then why does the motion pick up again? Why doesn't the spring just stay flat?

These pictures show the motion of wave pulses along a spring. To make a pulse, one end of the spring was shaken by hand. Movies were filmed, and a series of frames chosen to show the motion. (a) A pulse travels to the left. (b) Superposition of two colliding positive pulses. (c) Superposition of two colliding pulses, one positive and one negative. (PSSC Physics)

The medium is not transported with the wave.

As the wave pattern passes the rubber duck, the duck stays put. The water isn't moving with the wave.

The sequence of three photos above shows a series of water waves before it has reached a rubber duck (left), having just passed the duck (middle) and having progressed about a meter beyond the duck (right). The duck bobs around its initial position, but is not carried along with the wave. This shows that the water itself does not flow outward with the wave. If it did, we could empty one end of a swimming pool simply by kicking up waves! We must distinguish between the motion of the medium (water in this case) and the motion of the wave pattern through the medium. The medium vibrates; the wave progresses through space.

Self-Check

As the wave pulse goes by, the ribbon tied to the spring is not carried along. The motion of the wave pattern is to the right, but the medium (spring) is moving from side to side, not to the right. (PSSC Physics)

In the photos above, you can detect the side-to-side motion of the spring because the spring appears blurry. At a certain instant, represented by a single photo, how would you describe the motion of the different parts of the spring? Other than the flat parts, do any parts of the spring have zero velocity?

Answer

The leading edge is moving up, the trailing edge is moving down, and the top of the hump is motionless for one instant.

The incorrect belief that the medium moves with the wave is often reinforced by garbled secondhand knowledge of surfing. Anyone who has actually surfed knows that the front of the board pushes the water to the sides, creating a wake. If the water was moving along with the wave and the surfer, this wouldn't happen. The surfer is carried forward because forward is downhill, not because of any forward flow of the water. If the water was flowing forward, then a person floating in the water up to her neck would be carried along just as quickly as someone on a surfboard. In fact, it is even possible to surf down the back side of a wave, although the ride wouldn't last very long because the surfer and the wave would quickly part company.

A wave's velocity depends on the medium.

The wave pattern moves to the left while the earthworm moves to the right. The medium - the worm's body segments - does not move along with the wave pattern.

A material object can move with any velocity, and can be sped up or slowed down by a force that increases or decreases its kinetic energy. Not so with waves. The magnitude of a wave's velocity depends on the properties of the medium (and perhaps also on the shape of the wave, for certain types of waves). Sound waves travel at about 340 m/s in air, 1000 m/s in helium. If you kick up water waves in a pool, you will find that kicking harder makes waves that are taller (and therefore carry more energy), not faster. The sound waves from an exploding stick of dynamite carry a lot of energy, but are no faster than any other waves. In the following section we will give an example of the physical relationship between the wave speed and the properties of the medium.

Once a wave is created, the only reason its speed will change is if it enters a different medium or if the properties of the medium change. It is not so surprising that a change in medium can slow down a wave, but the reverse can also happen. A sound wave traveling through a helium balloon will slow down when it emerges into the air, but if it enters another balloon it will speed back up again! Similarly, water waves travel more quickly over deeper water, so a wave will slow down as it passes over an underwater ridge, but speed up again as it emerges into deeper water.

Hull speed.

Wave patterns

Circular and linear wave patterns, with velocity vectors shown at selected points.

If the magnitude of a wave's velocity vector is preordained, what about its direction? Waves spread out in all directions from every point on the disturbance that created them. If the disturbance is small, we may consider it as a single point, and in the case of water waves the resulting wave pattern is the familiar circular ripple. If, on the other hand, we lay a pole on the surface of the water and wiggle it up and down, we create a linear wave pattern. For a three-dimensional wave such as a sound wave, the analogous patterns would be spherical waves (visualize concentric spheres) and plane waves (visualize a series of pieces of paper, each separated from the next by the same gap).

Infinitely many patterns are possible, but linear or plane waves are often the simplest to analyze, because the velocity vector is in the same direction no matter what part of the wave we look at. Since all the velocity vectors are parallel to one another, the problem is effectively one-dimensional. Throughout this chapter and the next, we will restrict ourselves mainly to wave motion in one dimension, while not hesitating to broaden our horizons when it can be done without too much complication.

Discussion Questions

A	[see above]
B	Sketch two positive wave pulses on a string that are overlapping but not right on top of each other, and draw their superposition. Do the same for a positive pulse running into a negative pulse.
C	A traveling wave pulse is moving to the right on a string. Sketch the velocity vectors of the various parts of the string. Now do the same for a pulse moving to the left.
D	In a spherical sound wave spreading out from a point, how would the energy of the wave fall off with distance?

golphysics

2008-12-20T00:01:00.000-08:00

Work with gravity and bouncy wall changes as you try to bounce your ball all the way to the goal.
copyright 2008 Flash Physics Games | Contact: blognog [at] hotmail.com

neon layers

2008-12-19T00:01:00.000-08:00

Alternate through the neon layers until you safely get the glowing ball to the exit.
copyright 2008 Flash Physics Games | Contact: blognog [at] hotmail.com

Stack the police

2008-12-18T00:01:00.000-08:00

Stack the police as quickly as you can by using your trusty tractor beam.
copyright 2008 Flash Physics Games | Contact: blognog [at] hotmail.com

Swinging Ball

2008-12-17T00:01:00.000-08:00

Swing your way all of the way to the exit by using your friend, momentum.
copyright 2008 Flash Physics Games | Contact: blognog [at] hotmail.com

Boom Bot

2008-12-16T00:01:00.000-08:00

Get your bot to the exiting door by using highly explosive bombs.
copyright 2008 Flash Physics Games | Contact: blognog [at] hotmail.com

Deconstruction

2008-12-15T00:01:00.000-08:00

Destroy all of the buildings using explosives and avoid the rubble at all costs!
copyright 2008 Flash Physics Games | Contact: blognog [at] hotmail.com

Steer Wheels

2008-12-14T00:01:00.000-08:00

Use your steer wheels to guide the yellow ball until your reach the yellow landing zone.
copyright 2008 Flash Physics Games | Contact: blognog [at] hotmail.com

Ragdoll Cannon

2008-12-13T00:01:00.000-08:00

Get your ragdoll to the landing zone by shooting it out of a cannon.
copyright 2008 Flash Physics Games | Contact: blognog [at] hotmail.com

Totem Destroyer

2008-12-12T00:01:00.000-08:00

Carefully remove all of the totems until there are no more and get your idol to land on one of the black totems without letting it hit the ground.
copyright 2008 Flash Physics Games | Contact: blognog [at] hotmail.com

DUI

2008-12-11T00:01:00.000-08:00

Use your mouse to click on each of the falling blocks as quickly as you can.
copyright 2008 Flash Physics Games | Contact: blognog [at] hotmail.com

rag doll cannon 15

2008-12-10T00:01:00.000-08:00

In this sequel your mission stays the same, get your ragdoll to the landing zone by using the least amount of cannon shots possible.
copyright 2008 Flash Physics Games | Contact: blognog [at] hotmail.com

unstack

2008-12-09T00:01:00.000-08:00

matcheroo

2008-12-08T00:01:00.000-08:00

Physics Games | Flash Physics

2008-12-07T00:01:00.000-08:00

Physics Games --- Magic Pen

2008-12-06T00:01:00.000-08:00

One more, physics games is fun

copyright 2008 Flash Physics Games | Contact: blognog [at] hotmail.com

Physics Games --- Color Infection

2008-12-05T20:07:00.000-08:00

Game this physics was very exciting, we in asked to count on when a ball in gave an urging, to he could exact touched the ball that in the intention

copyright 2008 Flash Physics Games | Contact: blognog [at] hotmail.com

The monkey and the hunter

2008-12-02T00:02:00.000-08:00

From Physclips: Mechanics with animations and film.

How far does a projectile 'fall away' from the line along which it is aimed?

What happens if you aim at a target, and if the target begins to fall just as the projectile is launched?

The Monkey and the Hunter

thinking that he will thus fall below the trajectory of the bullet. (Monkeys don't study physics.) What happens?

Before doing the mathematics, let's look at the situation. The monkey falls and accelerates downwards at g. The bullet starts off travelling along the aiming line, but it is also accelerated downwards at g. So we can imagine its motion as 'falling below the aiming line'. At equal times, it will fall below this line by an amount equal to the distance fallen by the monkey. At the time when both have the same horizontal position, they will both have fallen the same distance. This is not good news for the monkey.

The aiming line or sighting line is the path taken by light, which is not affected by gravity (or at least not measurably affected by the Earth's gravity) and so is a straight line - the black line in this graph. If the projectile is fired along this line, its initial velocity, v₀, is along that line. If air resistance may be neglected, then there are no horizontal forces and so the horizontal component of velocity, v_x, is constant. The vertical component, on the other hand, is steadily decreased by the acceleration due to gravity. The resulting trajectory is shown at right.

The sequential positions of the bullet shown at right are equally spaced, 2 units apart. Because the horizontal component of the velocity v_x is constant, then these represent equally spaced times and the horizontal position x is just v_xt. The vertical lines between the aiming line and the trajectory show how far the bullet has fallen below the aiming line. The vertical component of velocity v_y has an initial value v_y0, but decreases due to gravity:

_y0

Integrating this with respect to time gives us the vertical position of the projectile:

₀

_y0

We can think of the aiming line as the path of an imaginary projectile that was not affected by gravity. This would be:

_aim

₀

_y0

The amount by which the projectile has fallen below the aiming line is

_aim

Dy is the height of the vertical red lines in the figure, which increase in the ratio 0, 1, 4, 9, . . . n².

Now if you look at the algebra above and put v₀ = 0, you will see that gt²/2 is the distance fallen in time t by an obect starting with zero vertical velocity. So the monkey falls as far below the aiming line as the bullet does.

No monkeys were hurt in the making of this clip

₀

₀₎

_x0

_y0

_x0

_y0

_x0

₀

_y0

_x0

_aim

_x0

_aim

Projectiles. Air resistance, gravity, range, trajectory.

2008-12-02T00:01:00.000-08:00

The hammer and feather experiment, on earth and on the moon

When do objects fall with the same acceleration?

this link

We return to look quantitatively at the effects of the earth's atmosphere below. Briefly, however, the air resistance will be small compared to the weight if:

A simple experiment to measure g

Five masses are attached to a string: one at the end, and one each at 15, 60, 135 and 240 cm. In other words they are at

1².L, 2².L, 3².L, and 4².L

So, in the experiment shown, their initial heights are proportional to the squares of the integers. The analysis below shows that, when they are dropped simultaneously by releasing the string, the constant acceleration hypothesis predicts equal intervals between their arrivals.

This makes a for a good, quantitative lecture demonstration. Here the room is carpeted, so we add a surface that makes a short clear sound on impact. We have two strings with masses: the other (not shown) has the masses equally spaced. When they are dropped, one shows crescendo, the other shows crescendo e accelerando.

How to measure the times? A cheap microphone, a computer sound card and a simple sound editing package gives reasonably good precision in time. Here we used the soundtrack from the video. In a classroom, however, I do it differently: I listen to the rhythm, which I think of as four quavers in a 2/4 bar, then I count the number of bars in 10 or 20 seconds. (I know a few conductors who wouldn't need the clock, but my internal metronome is not as well calibrated as theirs.)

Of course, since the 'zero' ball has no fall and no sound, the zero of time is not measured. For didactic purposes, this is an advantage: one can plot the results and take the slope of a graph.

As a lecture demonstration, this gives results with a scatter of about 1% around the accepted value. (The analysis shown in the graph is the first one that we did.) As a teaching laboratory exercise, performed carefully by two people rather than one, it would be easy to do better than this. Although g is usually only given to two significant figures, that second figure requires better than 1% accuracy.

Notice that the balls bounce, and that the sounds of their second landings are recorded as the smaller spikes on the sound track. Using the step button, one can also observe the increasing blur as the velocity increases. The exposure time is presumably constant, but the mass moves further during each successive exposure.

Some other simple physics experiments are described in Interesting and inexpensive experiments for high school physics.

The independence of vertical and horizontal motion

If you fired a launch a projectile horizontally and drop a similar projectile at the same time, which will land first? We expect any forces due to air to be nearly symmetric in the vertical direction. For billiard balls in the class room these forces will be small compared to the weight. This apparatus launches one ball horizontally and releases the other with only a small horizontal component of velocity. Two panels are placed on the floor to increase the audio signal of the arrival time.

Using the step button, one can compare the vertical positions of the two balls at each stage.

An interesting observation about this demonstration. When I do it in class, I usually stand on the ladder. For this shot, we decided that it looked neater if I stood out of the way of the two trajectories. This reduces greatly the inertia associated with the launch system. Consequently, there is a recoil of the launcher and ladder, which is clearly visible in the close up in the multimedia presentation, and which contributes to the horizontal velocity of the ball on the right. We considered reshooting it, but decided to retain this version, to use again when analysing collisions.

A simple range experiment

The grape rolls off a table, 77 cm high. Because it rolls horizontally on the table, its initial vertical component of Using the step button, we observe that the grape took 10 frames to travel 56 cm. The camera runs at 25 frames per second, so the horizontal component of velocity is 1.4&nbspm.s^-1, in agreement with the calculation. Note the angle of the rebound: because the grape rolled off the table, it was spinning when it hit the carpet. This caused it to accelerate to the right during the collision.

Trajectory of a projectile

If we neglect air resistance, then the acceleration is − g downwards: there is no horizontal acceleration. As we have seen above, horizontal and vertical motion are independent. So we can write the equations for motion in the vertical and horizontal motion as we did in the introductory module:

₀

Range as a function of launch angle

Suppose that you launch a projectile at 20 m/s, and that the launch speed is independent of the launch angle. Further, let's launch it from ground level.

How far will it go, and what angle gives the greatest range?

We choose the origin to be the position from which we are launching. So x₀ and y₀ are zero. The equation above now simplifies to:

Note, by the way, that the parabola is dotted below the x axis. This part of the curve is given by the same equation for the motion of a projectile, but below the axis our object is no longer a projectile: it is a buried, rather than being in free fall. It is rather common in physics that equations produced by applying a general principle to a particular case have only a limited range of applicability.

So, if we set x = R and y = 0 in the equation above, we have:

_y0

₀

A simple but useful introduction to calculus

So, should you throw a ball at 45° to achieve maximum range? No. First, we assumed above that launch speed was independent of angle. This is not the case for throwing: most people can throw considerably faster at low angles. There's one reason for throwing at a flatter angle.

Second, we have neglected air resistance. Although this is a small effect for low speeds and large objects, it is often non-negligible for balls, as we show below. For objects that, like balls, are symmetrical about the line of their velocity, the resistance always opposes the motion, so it means that the vertical component of velocity decreases more rapidly during ascent, and increases less rapidly during descent, than would be the case in a vacuum. Throughout the whole trajectory, it reduces the horizontal component of velocity. The consequences of this is that the ball falls short of the trajectory calculated by neglecting air resistance. For this reason too you should throw with θ less than 45°. (For a frisbee, a discus or a javelin, whose axes of symmetry are not parallel to the velocity, aerodynamic lift is often important.)

Drag and air resistance: getting quantitative

Near the surface of the earth, a falling object is acted on by its weight and by forces exerted by the air. Moving an object through the air at speed requires a force, because air must be accelerated. There may also be a force of buoyancy exerted by the air -- which is why hot air balloons and helium-filled airships are not projectiles!

Although this page is part of the kinematics section, this seems like a good place for a quantitative analysis, to which we can return later, after studying force and work. So, let's look at the air resisance or turbulent drag force.

This non-streamlined bus, with cross-sectional area A, is travelling at velocity v. What force is exerted by the air on the bus?

When an object moves through a fluid at sufficiently low speed, the drag force is due to viscosity, which gives rise to a force somewhat like friction. (Without going into viscosity here, honey has greater viscosity than water, which has greater viscosity than air.) At sufficiently high speeds, the drag is due to the acceleration of the nearby fluid, and the subsequent loss of kinetic energy by that fluid in turbulent flow. For macroscopic objects at ordinary speeds in air, the drag is almost entirely turbulent, and that is what we shall analyse.

In a time t, the bus travels vt. In that time, approximately all of the air in a volume A.vt must be accelerated to the speed of the bus. The density of the air is ρ, so the mass of this volume is ρ.Avt. So the air will gain a kinetic energy:

The work to produce this kinetic energy is done by the bus, which pushes the air forwards. From Newton's third law, the force that the bus exerts on the air equals in magnitude that which the air exerts on the bus. The latter force is called the drag, Force, F_drag. So the work done to accelerate the air is F.vt.

From the work-energy theorem, W = ΔK, so in this case

_drag

Cancelling the vt gives:

_drag

The "≈" came because we said that approximately all of the air in front of the bus is accelerated to match its speed. This is a bit complicated, because the air is pushed sideways, and some of it is accelerated to speeds less than v. Some air is drawn behind the bus, too. Further, it depends on how streamlined the object is, on both front and back.

There is no simple way to calculate the drag force. For a given object, it can be measured. From such a measurement, we define the drag coefficient, C_D thus:

C_D depends strongly on the geometry, It depends much less strongly on the size and speed of the objects. So one can measure C_D for an object in a wind tunnel, and use that value to estimate F_drag for other objects with similar shapes. For a sphere at ordinary speeds, the value is about 0.5. For streamlined automobiles it is about 0.3-0.4. For a cyclist it is a bit less than 1.

In the multimedia module, we referred to the force on a hand held out of a moving car. Taking just order-of-magnitude values, let's put 100 km/hr, CD ~ 1, and A ~ 0.01 m². The density of air is 1.2 kg.m⁻³. Substituting in the equation above, the force is several newtons.

When can one ignore air resistance?

For a grape, m is about 5 g and r about 10 mm. For an apple, m is about 100 g and r about 30 mm. For a sphere, the drag coefficient is about 0.5. The density of air is 1.2 kg.m⁻³. So, at a speed of 5 m.s⁻¹,f is approximately 0.05 for the grape and 0.02 for the apple. Dropped from the hand, the objects don't go much faster than this, and are slower for much of the trajectory. So the difference is small.

Balls, however, are thrown, hit and kicked much faster. Further, while some balls (cricket, golf) are dense, others (footballs) have low density. How important is drag? Why not do some examples?

A projectile question: When does the cricket ball travel its fastest (towards the batsman)?

This question was asked by Dan Trouw of Darwin. Here is the situation. The ball is released from the hand with the arm outstretched above the bowler's head, so it typically has a little more than two metres to fall to ground level. On the other hand, air resistance is slowing it from the moment it leaves the hand. Which force 'wins'?

The answer is not immediately obvious -- to me, at least. Although a cricket ball is dense, from observatoin, air resistance is not negligible: The trajectory of a well-hit ball is noticeably not a parabola: it falls more rapidly than one would expect. (My excuse for missing that important catch on the boundary, and I am sticking to it.).

As above,

F_drag = (1/2)C_DAρv²,

For a sphere C_D is typically about 0.5. So we have

F_drag ≈ 0.25πv²ρv²,

For a very fast bowler, with v = 40 m/s, this force is about 2 newtons. For such a bowler, the drag force is about the same as the weight of the cricket ball. For someone bowling at say 20 m/s, the drag force would be only a few tenths of the weight.

Thus, for a ball bowled vertically downwards (a new meaning for shooting one's self in the foot), and for a slow bowler, weight would be more important than drag, as it clearly is when the ball is dropped.

However, the ball is not bowled vertically down -- at least not deliberately -- and that is important for this question. The effect of direction can be included by considering kinetic energy. Gravity has only about two metres of downwards vertical travel over which to accelerate the ball, while the drag has about twelve metres (more for a slow bowler) to decelerate it. And a very slow bowler would have to bowl the ball slightly upwards ("give it a bit of air") just so that it covers the distance. In which case, gravity would act to slow it at first, not accelerate it.

Consequently, drag has a greater effect on reducing the kinetic energy (and thus slowing it) than has gravity in increasing it. So, even for quite slow bowlers, the fastest that the ball is travelling is just when it leaves the bowler's hand -- until, of course, it reaches the bat.

Graphs, errors, significant figures, dimensions and units

2008-12-01T00:01:00.000-08:00

The first module in Physclips uses displacement-time and velocity-time graphs for a man walking in a straight line, so we'll begin with this animation.

⁻²

Let's now see how the method of dimensions can be useful, via this

Example: how does the frequency, of a pendulum depend on the length?

We know that this depends on the length, L -- a long one swings more slowly than does a short one. It also depends on the strength g of the gravitational field -- it won't swing at all without one. Does it also depend on the mass, m? On the temperature, T? Let's write for the frequency, f:

⁻¹

⁻²

[T] ⇒ −1 = −2b

[L] ⇒ 0 = a + b

[M] ⇒ 0 = c

[Temperature] ⇒ 0 = d.

The other two equations tell us that b = 1/2, and that a + b = 0, so a = −a = −1/2. So, substituting in our original equation for the frequency,

^−1/2

^1/2

* I raise a couple of tiny caveats, to preempt the pedants. For a pendulum whose mass is comparable with the that of the planet upon which it is mounted, the pendulum mass does appear -- or at least the ratio of these two masses appears. Further, we have cheated a little on the temperature, because we could write temperature in units of energy. Doing so, the conversion factor would be Boltzmann's constant, whose very small size would give us the clue that temperature is only relevant in mechanics for objects of molecular size. And on this molecular scale we should often need to use quantum mechnics rather than Newtonian mechanics.

Other units

With rare exceptions, scientists use the SI system of units. (SI stands for Système International d'Unités.) This system is based on the kilogram for mass, the metre for length, the second for time, the ampere for electric current, the kelvin for temperature, the mole for chemical quantities and the candela for luminous intensity. Other systems are the British imperial system and natural units.

Physclips is a scientific presentation, and we use only the SI. If you enounter problems stated in other units, the simplest procedure is often to translate the problem into SI, solve it, then translate the back. This sounds like extra work, but it is usually much less than the extra work required in using the imperial system of units, which has internal conversion factors.

In the United States of America, Liberia and Myanmar, the British imperial system is the official system. This system used to be much more widespread, and vestiges of it remain in other countries that are in the process of 'going metric', ie converting to the SI.

Dealing with or converting from the imperial system usually involves just a multiplictaive factor. For instance, the inch, an imperial unit of length, is officially defined to be equal to 25.4 mm. These multiplications can become awkward in some cases: consider this imperial unit of thermal conductivity, one British Thermal Unit per second per square foot per degree Farenheit per inch. One can see why it exists, but it is ugly and inconvenient. (For comparison, the SI unit thermal conductivity is W.m⁻¹.K⁻¹.)

Some confusion arises, however, because of the different colloquial use of units of mass and force in the SI and imperial system. In the imperial system, the unit of force is the pound-force, or sometimes, as in many American physics text books, as just the pound. The unit of mass in the imperial system is the slug, which is a mass that is accelerated by one pound at one foot per second per second. The slug is 14.5939 kg. These equations, which are definitions, allow us to compare the units of mass and force:

⁻²

Imperial

⁻²

In imperial units, the gravitational acceleration is 32 feet.second⁻². Consequently, a slug weighs 32 pounds.

The slug is very rarely used. Pound is used colloquially as a unit of quantity -- a pound of apples colloquially means a quantity of apples that weighs a pound-force (at the earth's surface). There is another imperial unit of force, the poundal. This is defined as the force required to accelerate at one foot.second⁻² a mass whose weight is one pound. So a pound is 32 poundals.

The units mentioned above are related to features of the earth (its circumference originally determined the metre, and the second is related to the day) or of artifacts on earth, such as the standard kilogram, or of particular substances, especially water. The laws of physics and combinations of them yield natural units, which are used by some theoretical physicists, especially cosmologists. The speed of light, for instance, is taken as the unit for speed. Although this makes equations look simple these units are, in general, inconvenient for measurement. For instance, the natural units of length and time are inconveniently small (The Planck length is 1.6 x 10^-35 metres, the Planck time is 5.4 x 10^-44 seconds). See The Planck scale for more detail.

Calculus

2008-11-30T00:01:00.000-08:00

Calculus: differentials and integrals. A simple introduction from Physclips

Calculus is easier than you think. Here's a simple example: the bucket at right integrates the flow from the tap. The flow is the time derivative of the water in the bucket. The basic ideas are not more difficult than that. Calculus analyses things that change, and physics is much concerned with changes. For physics, you'll need at least some of the simplest and most important concepts from calculus. Fortunately, one can do a lot of introductory physics with just a few of the basic principles.

So stick with us: differentiation really is just subtracting and dividing, and integration really is just multiplying and adding. This short introduction is no substitute, however, for a good high school calculus course: we are going to take some short cuts of which mathematicians may disapprove.

Differentiation: How rapidly does something change?

The velocity is the rate of change of displacement. Let's look at a very simple case. The strange chap in this animation is moving in a straight line at a constant speed of one metre per second. This means that, for each second, his displacement from the starting position increases by 1 m for each second that he travels. (Velocity is a vector, meaning that it has direction as well as magnitude. So here we could say that his speed is 1 m/s but his velocity is 1 m/s towards the right. In these examples, we shall consider only motion in a straight line, so we shall omit the direction. Positive velocity will mean going to the right, negative velocity is travelling to the left. (By the way, the magnitude of velocity is called the speed, which we could write as |v|. Speed is a scalar, velocity is a

vector

When the clock strikes zero, he is at x = 3 m. We call this his initial displacement and write x₀ = 3 m.

When the clock reads t = 2 s, he is at x = 5 m. So what is v? Displacement has increased by 2 m, time has increased by 2 s, so v is

Note the geometrical significance of taking the derivative: looking at the triangle drawn on that graph, the height is 2 m, and the base is 2 s, so the derivative is the slope of the graph. Often it is said to be the rise (here Δx) over the run (here Δt). Note, too, that for this case, it doesn't matter how big or where we draw the triangle. If we made the run 0.3 s, the rise would be 0.3 m, and so on. That, however, is just for this special case, where v is constant, and so v and are the same. The derivative of x in this case is constant, as we show in the v(t) graph above.

Varying derivatives

What if v is not constant? Here's a simple case in which the same bloke is accelerating forwards, so his velocity is increasing. How can we work out his velocity at any particular time t, just from the graph x(t)? We want to find the slope of this curve, at different values of t, as is shown in the animation. How can we do it? Well, in most cases, and especially in the case of experimental measurement, we do the same thing that we did for the simple case: we draw a triangle and put change in displacement Δx over change in time Δt. Again, this is numerical differentiation.

If we want the slope at t = 0 s, we might take t = 0 s and t + Δt = 2 s. So Δt is two seconds, using the notation introduced above. But the trouble is that that gives us a triangle whose slope (the black line) is somewhat greater than the slope at t = 0 s (the red line). In fact, the slope of the black line gives us the average velocity between 0 and 2 seconds, but that is not what we want. We can do better, in this case at least, by drawing our triangle on either side of t = 0 s. Try it, but you'll find that that gives an overestimate, too.

We can do better by taking a smaller value of Δt. In principle, we might expect the estimate to improve as we take smaller and smaller Δt. Here arises a practical problem: If we are measuring the values, there is inevitably a limit to the precision of each measurement and, as the Δt and Δx get smaller, this error becomes proportionately more important. If we are calculating x and t numerically, we face the same problem. So, in practice, we make a compromise on the size of Δt: small enough to give the local shape of the curve but large enough that the measurement or calculation errors are small. There are other tricks that we'll see below.

Analytical derivatives

But what if we 'know' the formula for the function x(t)? I have put 'know' in quotation marks, because for anything in physics, the only things that we know are the measurements. There are only a finite number of these, so we just have a set of points on a graph. What we can do is to find a mathematical model, a formula that goes close to the points on the graph. For example, if the acceleration is constant, we might use x = x₀ + v₀t + (1/2)at², as we do in this module. We can now choose whatever t we like, and calculate x to whatever precision we need, though of course the final precision will depend on how well we know x₀, v₀ and a, so we are still limited by measurement.

Power terms and polynomials

Let's have a look at these terms in turn. If x = x₀, then it's simple: no variation in x, so the derivative would be zero. What about x = v₀t, in other words constant velocity? This is like the first example we did: the derivative is constant, and it equals v₀. So, the derivative of a constant is zero, and the derivative of a term that is proportional to t is just the constant of proportionality or, in standard terms, the coefficient of t.

But what of a term like x = Ct²? (In our example, C = a/2, but let's keep it general.) Let's make the 'run' for our slope calculation from t to (t+Δt). Then the 'rise' on the triangle will be from Ct² to C(t+Δt)². So the slope is

Now let's make Δt and Δx extremely small, and we signify this by writing them as dt and dx.

As we've said, dt is very small, and can be made smaller than anything that we could measure. So we can neglect it on the right hand side. We don't neglect it on the left hand side, because there we have the ratio of two small things, and that ratio need not be small. So here we have one useful case for taking derivatives: the derivative of Ct² is just 2Ct.

At this stage, if you were doing this in a mathematics school, you might start to worry about infinitesimals, and taking the limit as Δt goes to zero. I may get into trouble for pointing this out, but the universe doesn't have infinitesimals, and quantities don't go to zero in physics. Infinitesimals, like many things in mathematics, are human inventions. So, for most purposes in physics, the limit taken is just the size necessary to have mathematical precision greater than that of our measurements, or greater than that of our numerical calculation. (You really should take that mathematics course, but you won't need infinitesimals in physics.)

Let's summarise what we have so far:

Let's graph these, setting the constants equal to one. We'll also omit units on the axes because, although you may find it helpful to think of the vertical axis as displacement and t as time to give a concrete example, the results are general. For that reason, we'll use y as the vertical axis from here on.

In all of the graphs on this page, the red curve is the derivative of the purple one. It is a good exercise to compare the two, and to check that, in all cases and over the whole curve, that the red line represents the slope of the purple one. Here, for instance, the slope of y = 1 is zero. The slope of the straight line y = t is obviously consant. In the third curve, you can see that the slope is negative at first, zero at t = 0, and then becomes increasingly positive.

Perhaps you see a pattern here? Taking a positive integer n, and expanding (t + Δt)ⁿ, you'll see that, if y = Ctⁿ, the derivative of y is nCt^n-1. This is often written thus:

This result is more general: it actually holds for all real values of n, positive and negative. n = 0 is a special case, because it gives a factor of zero on the right hand side. This will be important when we come to look at integrals, below.

Sums of terms

If x is increasing with rate (dx/dt) and y is increasing with rate (dy/dt), what is the rate of increase in (x+y)? This is an easy one: if, over a certain time t+ Δt, x increases by Δx and y increases by Δy, then the change in (x+y) is (Δx+Δy).

The rate of change in the sum of functions is equal to the sum of their individual rates of change. (The derivative of the sum is the sum of the derivatives.) With this unsurprising result, we can now differentiate polynomials, such as this: If

y = A + Bt + Ct² Dt³, then
dy/dt = B + 2Ct + 3Dt².

Trigonometric functions

vectors

The definition of sine of an angle uses a right angled triangle. It is the ratio of the side opposite the angle to the hypotenuse of the triangle. The definition of cosine is the other side (that adjacent to the angle) divided by the hypotenuse.

In this diagram, first look at the triangle with blue sides. The radius is drawn at the angle θ. The vertical side is, by definition, sin θ and the horizontal side is cos θ.

Now let's change θ by an amount Δθ: we get another radius and another triangle, shown here with two of its sides in green. By how much have we changed sin θ? In other words, what is Δ(sin θ)?

By definition, Δ(sin θ) is the amount that in θ increases when θ increases by Δθ). So it must be (sin(θ+Δθ) - sin θ). On this diagram, Δ(sin θ) is shown as the rise of the small triangle, shown in red. (For clarity, this triangle is repeated outside the main diagram.) The run of this triangle is -Δ(cos θ). The run goes to the left in this case, so it has a negative sign. This negative sign arises because, as θ increases in this quadrant, cos θ decreases.

Now look at the small right triangle in red. We call its hypotenuse h. Applying the definition of sine and cos, and remembering that the cos θ is decreasing, we have

Again, we use the convention that dx means a value of Δx small enough to make the approximations better than our measurements. So, in this limit, and using the results of the previous paragraph (θ = φ and h = dθ for small enough values of Δθ), the equations above become

d(sin θ) = dθ cos θ and d(cos θ) = − dθ sin θ

* Note the use of the definition: angle in radians = arc/radius. Consequently, the dθ in these equations must be expressed in radians. To convert degrees into radians, multiply by π and divide by 180°.

The chain rule

Suppose we have a function z that depends on t, in a way that allows us to calculate z if we know t. And suppose that x depends on z in a similarly explicit way. Just by cancelling a factor, we can write:

If all of these are very small quantities, then we write

This is the chain rule of differentiation, which we use when analysing circular and simple harmonic motion.

Circular and Simple Harmonic Motion

In both these cases, we consider uniform circular motion, in which the angle θ increases linearly with time, so (dθ/dt) is a constant, called the angular velocity ω. We can write this as

θ = ωt + φ

where ω is constant and where φ is the initial value of θ. We therefore write for the vertical displacement

y = A.sin θ = A.sin (ωt + φ),

When we take the time derivative of θ, we have dθ/dt = ω. So the chain rule gives:

v_y = dy/dt = (dy/dθ).(dθ/dt) = (A.cos θ).(ω) = A.ω.cos (ωt + φ).

So much for maths: where, physically, does the extra factor ω come from? If we double ω, we double the rate at which the angle is increasing in circular motion or the phase is increasing in simple harmonic motion. So the time for one complete circle or cycle is halved. If the displacement goes through the same variation in half the time, then the velocity is doubled. (Differentiating one more time gives an acceleration that includes, in both cases, a factor ω². So, if we double ω, the range variation in velocity is twice as great, and we go through that range twice as often, so the acceleration has an extra factor of 4.)

Integration: How do the results of a variable rate add up?

Let's say that the bucket already has in it a volume V₀ of water and that we put the bucket under the tap (we start integrating) at time t = 0. The tap is already on, with a flow rate f₀, called the initial flow rate. Consider the first short interval Δt. The flow rate is by definition the volume per second so, in the first time Δt, a volume of approximately f₀.Δt pours into the bucket. f₀.Δt is shown by the first red rectangle on the top graph. "Approximately" is there because, during Δt, the flow is not constant, but varying -- in our example it is increasing.

Similarly, at any time later, the volume going into the bucket in the short interval Δt is approximately f.Δt. So, for each Δt we add a step of f.Δt to the height of the purple curve representing the volume in the bucket, as shown in the lower curve. Notice that, when the flow is high, the area f.Δt is large, so that is when the volume in the bucket increases rapidly. Note that, when the flow falls to zero (tap off), the volume is no longer increasing. And of course a fall in the V(t) curve would mean water flowing out of the bucket, which we should call negative flow into the bucket. (For example, the bucket might have a leak.)

What is the volume V_f in the bucket at a final time t_f? (The subscript f here stands for 'final', not flow.) We can find the approximate volume as

₀

₁

₂

The "approximately" appears because the flow varies over the time Δt. However, if we make Δt small enough, this variation becomes smaller than the limit of our precision*. As above, when Δt is small enough, we call it dt. So the equation above becomes:

₀

₁

₂

That sum is called the integral of f with respect to t

limits of integration

In the way I've presented this example, V₀ is a constant of integration. Integration doesn't tell you the complete answer, it only tells you how much something has changed during the process. In this case, to know the final volume in the bucket, we need to know not only the integral of the flow, but also how much was in the bucket before it started to integrate the flow. In most cases, you will need to find the constant of integration -- very often by using the initial conditions, as we did here.

Perhaps now is a good time to go back to the animations above and check that integrating the velocity (finding the area under the curve) gives the displacement.

Numerical integration

If we had a set of numerical values for f(t) -- whether experimental values or values calculated for a given mathematical function -- then we could integrate just as described above: mutliple f by Δt and add. That's what my pocket calculator does when I hit the "∫" key. You've probably thought that it would be smart to space Δt either side of the time whose value gives you f, and you should remember that next time you have to do a numerical integration.

A very important practical point: when we made Δt small in differentiation, we encountered the problem that the ratio Δx/Δt became sensitive to experimental or computational error as Δt became small. This problem does not arise in multiplication. Even better, the computation errors, being sometimes positive and sometimes negative, tend to cancel out. So numerical integration is much easier and safer than numerical differentiation. The latter requires considerable caution (and is why my calculator doesn't have a "differentiate" button).

Analytical integration

This section might be shorter than you expect. We've mentioned above that integration is the opposite to differentiation:

The rate at which something changes is its derivative, but
You can recover that something by integrating the rate at which it changes.

In differentiation, we subtract V(t) from V(t+dt) and divide by dt
In integration we multiply f(t) by dt and add it to V(t), where f is the derivative of V with respect to t.

So for analytical integration, we can use in reverse the tricks we established above for differentiation. Omitting constants of integration, we write

The derivative of tⁿis nt^n-1, so
The integral of nt^n-1 is tⁿ

provided that n is not equal to zero, because in that case the first equation gives us no information.

Dividing the last equation by n, and substituting m = n−1 gives

The integral of t^m is t^m+1/(m+1)

when m is not equal to −1.

1/x and the log function

What is a logarithm? A brief introduction

So, in this plot, the slope of the purple curve is given by the red curve (differentiation finds the rate at which it is increasing). The purple curve rises at a rate given by the red curve (integration of a function means successively adding its value).

For this and the next section, the example will use numbers x and y, rather than displacements as a function of time. Quantities with dimensions add extra constants, as we see below, and it is easier to begin without them.

The integral of sine and cos

Above we saw that the derivative of sin θ is cos θ, and the derivative of cos θ is − sin θ. So

The integral of cosθ is sin θ, and
the integral of sin θ is − cos θ.

The exponential function

The function e ^x is chosen and the value of e defined so that the derivative of e ^x is e ^x. In other words, e ^x is a curve whose slope equals its values at all points. So it is also its own integral.

On the graph, the curve (purple) shows e ^x vs x. In this the derivative is not shown in red, because the function and its derivative are equal. The straight line (green) is y = e .x. At t = 1, the slope of the curve is e ¹ = e . This is also the value of y at x = 1. Notice that, at x= 0, the slope of the curve is one. At x = −1, the slope is 1/e , etc.

Using the chain rule tells us that

The derivative of e ^ax is a.e ^ax, where a is a constant, and
The integral of e ^ax is e ^ax/a.

dimensions

₀

^−t/τ

₀

This example suggests a nice problem: what is the relation ship between force and velocity for a particle of mass m whose velocity is given by

₀

^−t/τ

₀

Techniques for integration

You need to assemble a little collection of integrals of simple functions, including those listed immediately above. There is a range of techniques for integrating more complicated expressions. In approximate order of frequency, these are the techniques I use:

Change the variables to make it more closely resemble an integral I do know. Here are a couple of examples:
- to integrate dx/(1+x), try the substitution y = 1 + x. dy here is equal to dx, so this gives an integral we've seen above.
- to integrate x.dx(1 − x²)^−1/2, try the subsitution x = sin θ.
Integrate by parts. This technique comes from the derivative of the product of two functions. (This is getting beyond what we need for the material here, so see a book on calculus for more details.)
Look up a table of integrals! These have pages and pages of integrals, which are presumably assembled "the other way round" -- ie the authors probably make a table of derivatives, then index it in terms of the integrals.
Try an algebraic mathematical package
Integrate numerically -- for many problems, this is the only way.

Go back to Physclips: mechanics with animations and video film clips

* You might ask what would happen if a quantity varied infinitely rapidly. However, physical quantities do not, so that's another thing we'll leave to the mathematicians.

What is a logarithm? A brief introduction.

First let's look at exponents. If we write 10² or 10³ , we mean

10² = 10*10 = 100 and 10³ = 10*10*10 = 1000.

So the exponent (2 or 3 in our example) tells us how many times to multiply the base (10 in our example) by itself. For this page, we only need logarithms to base 10, so that's all we'll discuss. In these examples, 2 is the log of 100, and 3 is the log of 1000. If we multiply ten by itself only once, we get 10, so 1 is the log of 10, or in other words

10¹ = 10.

We can also have negative logarithms. When we write 10^-2 we mean 0.01, which is 1/100, so

10^-n = 1/10ⁿ

Let's go one step more complicated. Let's work out the value of (10²)³. This is easy enough to do, one step at a time:

(10²)³ = (100)³ = 100*100*100 = 1,000,000 = 10⁶.

By writing it out, you should convince yourself that, for any whole numbers n and m,

(10ⁿ)^m = 10^nm.

But what if n is not a whole number? Since the rules we have used so far don't tell us what this would mean, we can define it to mean what we like, but we should choose our definition so that it is consistent. The definition of the logarithm of a number a (to base 10) is this:

^{log a}

In other words, the log of the number a is the power to which you must raise 10 to get the number a. For an example of a number whose log is not a whole number, let's consider the square root of 10, which is 3.1623..., in other words 3.1623² = 10. Using our definition above, we can write this as

^{log 3.1623}

However, using our rule that (10ⁿ)^m = 10^nm, we see that in this case log 3.1623*2 = 1, so the log of 3.1623... is 1/2. The square root of 10 is 10^0.5. Now there are a couple of questions: how do we calculate logs? and Can we be sure that all real numbers greater than zero have real logs? We leave these to mathematicians (who, by the way, would be happy to give you a more rigorous treatment of exponents that this superficial account).

A few other important examples are worth noting. 10⁰ would have the property that, no matter how many times you multiplied it by itself, it would never get as large as 10. Further, no matter how many times you divided it into 1, you would never get as small as 1/10. Using our (10ⁿ)^m = 10^nm rule, you will see that 10⁰ = 1 satisfies this, so the log of one is zero.

In physics, we normally use natural logs. Consider the function y = a^x, with a > 0. The larger the magnitude of a, the more steeply this quantity increases. The number e = 2.718... has the property that the derivative of e^x is just e^x. This function is plotted above. Logs to base e are usually written as ln so, for instance, we may write:

ln (2.718) = 1.000.

log₁₀x vs x.

Vectors

2008-11-29T00:01:00.000-08:00

A vector has magnitude and direction

Let's start with displacement, which is a vector. That means that it is specified by both a magnitude and a direction. If I move my bag 10 metres North, it is not in the same position as it would be if I were to displace it 10 metres East -- I'd certainly notice the difference when I went to look for my bag.

The magnitude of the displacement is just how far the object is from the origin of reference -- 10 m in my examples above. The direction may be specified in any convenient way. Here are some examples of vectors:

10 metres North (a displacement)
15 kilometres per hour, directly towards the opponent's goal (a velocity), and
9.8 m.s-2 down (an acceleration).

Directions are sometimes expressed in terms of North and South, East and West, up and down. We might also use "radially outwards from the centre of the circle" or "parallel to the initial direction" etc. In very many physics problems, we define a system of axes and use them.

An example

Let's consider a displacement in the x,y plane. Suppose I displace an object from the origin (0,0), to the point (2.0,1.0), where the units are metres. The purple arrow shows this displacement. We'll call this displacement r.

To distinguish a vector, some books used bold face. In handwriting, we usually use underlining. Here we use both. The symbol r is commonly used for a displacement, just as v is used for velocity and a is used for acceleration.

Magnitude and direction: How far have we moved it? From Pythagoras' theorem, the distance moved is ((1.0 m)² + 2.0 m)²)^1/2 = 2.2 m. In what direction have we moved it? We could describe this in several ways, including this: at angle θ from the x axis (in the positive mathematical sense). From trigonometry, tan θ = 1.0/2.0 so θ = 27°. In other words, we could describe it as

Note that, in both cases, we have given a magnitude and direction. This is necessary: because r is a vector, it has magnitude and direction. There are two pieces of informtion on the left hand side of the equation and so must there be on the right.

To write the magnitude of a vector, we simply use normal type face. In this example, note carefully the difference

r = 2.2 metres,

Components

To obtain the magnitude and angle from the components, we can use Pythagoras' theorem and the definition of the tan function, respectively, to give us:

^1/2

^-1

Unit vectors

If we say "Go 2.0 metres East, then 1.0 metre North", what does "East" mean? It tells us just direction, so does it have a magnitude? Suppose we give "East" a magnitude of one (not one metre, just one). Now we can treat "2.0 metres East" as a product. In this sense, East is a unit vector.

Of course, we don't need to use unit vectors in all circumstances. We can say "in the x direction" instead of i. However, apart from brevity, the advantage of unit vectors is that we can now do vector algebra very simply, as we shall see in the sections on addition, subtraction and multiplication of vectors.

Incidentally, you may also see unit vectors in circular, cylindrical or spherical polar coordinates. These include a unit vector in the radial or r direction and a unit vector in the θ direction. However, we don't use these in Physclips.

Vectors in three dimensions

In an (x,y,z) coordinate system, we shall use all three unit vectors of the Cartesian system and write, for a vector a:

There are two different ways in which the positive z axis could be at right angles to the postive x and y axes. By convention, we choose what is called the right-handed system: If your right thumb and index finger point in the direction of x and y, then it is possible to point your right middle finger in the direction of z, as shown in the sketch. (Apologies for the sketch: I'm right handed.)

A useful result comes from an extension of Pythagoras' theorem to three dimensions, which is also shown in that diagram. For the vector a, what is the magnitude a?

First, we can use Pythagoras for the hypotenuse h and write:

Now look at the triangle that has a as its hypotenuse. For this triangle,

Substitute for h from the first of these two equations into the second and we have Pythagoras in three dimensions, and thus means of calculating the magnitude of a vector with three Cartesian components:

Vector addition and subtraction

Adding vectors

First note that, on this diagram, two arrows are labelled b. However, each arrow has the same magnitude and the same direction, therefore they are the same vector. Now let's think of vectors a and b in this diagram as displacements. Let's say I move through the displacement a (ie, I travel from the tail of a to its head). To add a displacement b to this, I shall move through b, starting from where I have just arrived, at the head of a. In other words, I put the two vectors head to tail. So, in this diagram,

The order doesn't matter: if I duplicate a and put its tail at the head of b, I also get c, as you can see by mentally completing the parallelogram. So let's note that

which is not as trivial as it seems.

To quantify the addition, one could use geometry. However, if we know the components (or if we can find them) it's often easier to write:

c_xi + c_yj = (a_x + b_x)i + (a_y + b_y)j.

Now the last equation gives us an expression for the x component of c -- and also for its y component. In other words a vector equation in two dimensions gives us two scalar equations. c = a + b gives us:

c_y = a_y + b_y.

This is important to remember when you are 'chasing an extra equation' -- ie when you think that you have only n equations to solve for n+1 unknowns! And of course in three dimensions a vector equation gives three scalar equations.

Subtracting vectors

Let's take our previous example and rearrange

to obtain

We expect that b − b = 0. In other words, after a displacement b, − b must bring us back to where we started. So the vector − b has the same magnitude as b, but the opposite direction. In this diagram, we have therefore drawn

It is also helpful to look at these two diagrams and to remind yourself that (c − b) is the vector that you must add to b in order to get c. In other words, it is the vector a of the previous diagram.

Examples. Relative velocity is a good source of applications of vector addition and subtraction, so let's do some here.

What does a head wind feel like? Suppose that the wind is coming from the East (ie towards the West), at 5 m.s⁻¹. So the velocity of the wind, relative to the ground, is v_w = 5 m.s⁻¹ West. If you are bicycling East at 10 m.s⁻¹ (ie v_b = 10 m.s⁻¹ East), what will be the velocity of the wind, relative to you? How strong is the wind that you feel on your face?

The velocity of the wind with respect to the ground, v_w is usually called the true wind, and the velocity relative to an observer moving over the ground is called the apparent wind, v_aw. So this question would often be expressed thus: what is the apparent wind?

Here, it's easy: the wind is travelling at v_aw with respect to you, and you are travelling at v_b with respect to the ground, so the true wind is:

_aw

In the case of the cyclist going East and the wind going West, we could write the wind's velocity as

⁻¹

so the cyclist has a head wind of 15 m.s⁻¹ -- an answer that you probably did in your head already.

What does a cross wind feel like? What if you are travelling North, with the wind from the East, with the same speeds as before? What will be the apparent wind then?

The same colour coding is used as in the previous example. Cyclists often say "Tailwinds are rare, and downhill tailwinds don't happen." Using vector subraction, can you explain this belief?

Addition and subtraction of velocities is also very important to sailors, first because the velocity of the boat over the water adds to that of the water over the ground to give the boat's velocity with respect to the ground. More importantly, the relative velocity of the wind, as measured on the boat, is the 'true wind' (ie the velocity of the wind with respect to the land) minus the velocity of the boat with respect to the land. See The physics of sailing for examples.

This brings up the topic of relativity, in both Galilean and Einsteinien form. Relative velocities and relative accelerations are important in mechanics, so we discuss them here in relation to Newton's laws.

The scalar product (the dot product)

What does the product of two vectors mean? It must obviously be rather different to the product of two numbers. The answer is that it could mean anything that we define it to mean, provided that the definition is consistent. There would be no point in defining the product unless it were useful, so let's see where we could use it.

W = F.s.cos θ.

define the scalar product or the dot product of two vectors thus

a.b

a.a

ie the scalar product of a vector with itself is its magnitude squared. If we take the dot product of two vectors at right angles to each other, we encounter the sine of a right angle, which is zero. For reasons that will become clear below, we should note an important result that is less trivial than it looks:

a.b

b.a

We can apply these observations immediately to unit vectors (their magnitude is one, remember), to obtain these important results:

i.i

j.j

k.k

i.j = j.k = k.i = 0.

This allows us to simplify algebra somewhat. Take the two vectors

We can now write their scalar product in two different ways:

a.b

Setting these last two results equal, by the way, gives a really neat way of working out the angle between two vectors. You can also use the dot product to calculate the components of vectors, because, as you can see from applying the results above,

The vector product (the cross product)

We've just seen that the scalar product (or dot product) of two vectors was a scalar. The vector product (or cross product) is -- you've guessed already. First, here are a couple of examples where we need it.

electric motors

physclips main menu

define the vector product

a X b is pronounced "a cross b".

Thumb

Index

Middle

North

East

Down

any vector crossed with itself equals zero

Now for a surprise. To verify the mnemonic above, put your right thumb in the North direction and your forefinger in the East direction. Yes, North X East is indeed Down. Now put your right thumb in the East direction and your forefinger in the North direction. You have now seen that East X North is Up. In other words:

X b = − b X a,

We'll get some practice with that result, because, looking again at the unit vectors, we can write some very useful equations.

|i X i| = 1*1 sin 0 = 0. So:

i X i = j X j = k X k = 0.

Looking at the directions, we can also see that:

but notice that

a X b = (a_xi + a_yj + a_z k) X (b_xi + b_yj + b_z k)

The symmetry of this expression gives a mnemonic for writing it down, which is easily seen if you write down the symbols in this array:

a_x

a_x a_y a_z a_x

i j k i

Physclips home page

electric motors and generators

Sound Waves

2008-11-28T00:01:00.000-08:00

by Ron Kurtus (revised 26 June 2005)

Sound is a series of compression waves that moves through air or other materials. These sound waves are created by the vibration of some object, like a radio loudspeaker. The waves are detected when they cause a detector to vibrate. Your eardrum vibrates from sound waves to allow you to sense them. Sound has the standard characteristics of any waveform.

Questions you may have include:

What is sound?
What are the characteristics of sound?
How is sound created and detected?

This lesson will answer those questions. There is a mini-quiz near the end of the lesson.

Useful tools: Metric-English Conversion | Scientific Calculator.

Sound is waveform in matter

Sound is a waveform that travels through matter. Although it is commonly associated in air, sound will readily travel through many materials such as water and steel. Some insulating materials absorb much of the sound waves, preventing the waves from penetrating the material.

Does not travel in vacuum

Because sound is the vibration of matter, it does not travel through a vacuum or in outer space. When you see movies or TV shows about battles in outer space, you should only be able to see an explosion but not hear it. The sounds are added for dramatic effect.

Some atoms in space

Note that in outer space, there are actually some widely-spaced atoms and molecules floating around. But since they are so far apart, regular wave motion would not be great enough to detect.

Sound waves different than light waves

Also note that light and radio waves are electromagnetic waves. They are completely different than sound, which is vibration of matter. Electromagnetic waves are related to electrical and magnetic fields and readily travel through space.

Sound is a compression wave

The back-and-forth vibration of an object creates the compression waves of sound. The motions of a loudspeaker cone, drumhead and guitar string are good examples of vibration that cause compression waves. This is different than the up and down or transverse motion of a water wave.

(See General Wave Motion for more information.)

Transverse Wave (water wave)

Compression Wave (sound)

The illustration above shows a comparison of a transverse wave such as a water wave and the compression wave sound wave.

Characteristics of sound

A sound wave has characteristics just like any other type of wave, including amplitude, velocity, wavelength and frequency.

Amplitude

The amplitude of a sound wave is the same thing as its loudness. Since sound is a compression wave, its loudness or amplitude would correspond to how much the wave is compressed. It is sometimes called pressure amplitude.

Decibel

A common measurement of loudness is the decibel (dB). It is really 1/10 of a bel, which was named after the inventor of the telephone, Alexander Graham Bell. It is a complex unit that varies as the ratio of the logarithms of loudness.

Decrease in loudness

A sound wave will spread out after it leaves its source, decreasing its amplitude or loudness. The amplitude decreases as the square of the distance from the source. Also, if there is some absorption in the material, the loudness of the sound will decrease as it moves through the substance.

Speed or velocity of sound

The speed or velocity of sound in air is approximately 344 meters/second, 1130 feet/sec. or 770 miles per hour at room temperature of 20^oC (70^oF). The speed varies with the temperature of air, such that sound travels slower at higher altitudes or on cold days.

Note: The difference between speed and velocity is that velocity usually includes direction the of travel. We'll interchange them here, but in some cases the distinction is important.

A jet plane traveling at the speed of sound would be moving at about 680 mph at sea level. At very high altitudes, the speed required would be much lower.

(See Speed of Sound in a Gas for more information.)

Wavelength

Wavelength is the distance from one crest to another of a wave. Since sound is a compression wave, the wavelength is the distance between maximum compressions.

Frequency

The frequency of sound is the rate at which the waves pass a given point. It is also the rate at which a guitar string or a loud speaker vibrates. Frequency is also called the pitch of a sound. It is called the note in musical sounds.

Relationship

The relationship between velocity, wavelength and frequency is:

velocity = wavelength x frequency

Since the velocity of sound is approximately the same for all wavelengths, frequency is often used to better describe the effects of the different wavelengths.

Pitch

The pitch or note of a sound that we experience is determined by its wavelength or its frequency. The shorter the wavelength, the higher the frequency becomes, and the higher the pitch that we hear.

Creating and detecting sounds

Creating and detecting sounds are similar effects, but opposite. They demonstrate the duality of nature.

Creating sound

Whenever an object in air vibrates, it causes compression waves in the air. These waves move away from the object as sound. There are many forms of the vibration, some not so obvious.

The back and forth movement of a loudspeaker cone, guitar string or drum head result in compression waves of sound. When you speak, your vocal cords also vibrate, creating sound.

Blowing across a bottle top can also create sound. In this case, the air inside the bottle goes in a circular motion, resulting in sound waves being formed. Wind blowing through trees can also create sound this indirect way.

Sound can also be created by vibrating an object in a liquid such as water or in a solid such as iron. A train rolling on a steel railroad track will create a sound wave that travels through the tracks. They will then vibrate, creating sound in air that you can hear, while the train may be a great distance away.

Detecting sound

When a sound wave strikes an object, it can cause the object to vibrate. This leads to the method to detect sound, which requires changing that vibration into some other type of signal--usually electrical.

The main way you detect or sense sounds is through your ears. The sound waves vibrate your ear drum, which goes to the inner ear and is changed to nerve signals you can sense.

You can also feel sounds. Stand in front of a stereo or hi-fi loudspeaker on at full volume, and you can feel some of the vibrations from the music.

There are mechanical devices that detect sounds, such as the microphone. The sound vibrates a membrane, which creates an electric signal that is amplified and recorded.

(Also see Detecting Sounds for more information.)

Summary

Sound is a compression waveform created by the vibration of some object. Sound moves through air or other materials. The characteristics of sound are that it has amplitude, wavelength, frequency and velocity. Sound must cause another object to vibrate to be detected.

The Bernoulli Ball

2008-11-27T00:01:00.001-08:00

You may have seen beachballs balancing on vacuum cleaner exhaust hoses. You can make a smaller version by balancing a ping-pong ball on the stream of air from a blow dryer.

The toy we will make in this section uses the same principles as the beachball and the ping-pong ball. A small ball of balsa wood floats on a stream of air you blow through a bent tube. The ball has a small wire hook stuck through it. The object of the game is to hook the ball on a hoop attached to the tube.

The completed toy looks like this when made of clear plastic tubing:

The toy can be made of wood, with a large long hole drilled almost all the way through the length, and a smaller hole drilled near one end to meet the larger hole. In both versions, the idea is to make a tube that bends at a right angle, so the air coming from your mouth comes out vertically about six inches from your face.

The hoop is a loop of wire (I used brass because it looks nice) formed by bending the wire in half around a broomstick and then twisting the ends to form the handle of the loop. This handle is inserted into a hole drilled near the far end of the tube.

In the version shown in the photograph, I started with a clear plastic tube 1/2 inch in diameter. These can be found in hobby stores or hardware stores. I found mine in a specialty plastics store.

A smaller clear plastic tube about an inch high and 1/8 inch diameter is glued to the larger tube 1/2 inch from the end. It is glued over a hole drilled in the larger tube.

The far end of the tube is closed by a circle of plastic cut to fit the end of the tube and glued on.

The ball is made of balsa wood, but any lightweight material will do, such as a hollow plastic ball, or smooth styrofoam. It is a little bigger than half an inch in diameter, although the diameter is not critical.

A stiff wire hook is stuck through the ball, so that about an inch and a half sticks out the bottom, and an inch sticks out the top. After bending the top into a hook, it is about a half inch above the ball.

I made my hook out of stiff steel wire. It would be better to use something that doesn't rust, such as stainless steel, or brass. You can see the rust stains on the plastic tube in my version, caused by warm moist air.

To operate the toy, place the straight wire of the ball in the upright tube as shown in the first photograph. Take a very deep breath (you'll need all you can get). Blow through the open end of the large tube hard enough to raise the ball up to the level of the hoop.

The ball is now dancing around on top of the stream of air, and is very erratic. The trick is to control the height of the ball so when the hook is over the hoop, you can let the ball down gently, with the hook caught in the hoop. This is not easy, and usually takes several breaths. It is fun to watch the expressions on faces as people try to hook the loop on their last bit of air.

Why does it do that?

How can a ball balance on a stream of air? How can an airplane wing keep an airplane from falling? How can a sailboat sail into the wind? To explain these things, we have come up with elaborate physics and mathematical models such as the Navier-Stokes equations, and the Bernoulli principle.

These models use concepts such as pressure, velocity, density, and viscosity. They are not simple to explain or understand. However, all of the concepts used in these complicated models can be explained by thinking of interactions of molecules.

To explain what is happening with the balancing ball, it is useful to look at other, sometimes simpler experiments, and see what is going on there.

Some simple experiments

When you hold your hand out the window of a moving car, you can make the wind lift your hand by tilting your hand at an angle to the wind. Notice that the force you feel that lifts your hand seems to press on the bottom of your hand. If there is any force acting on the top of your hand trying to suck it upwards, it is definitely smaller than the force on the bottom of your hand pushing it upwards. [Try sucking on the back of your hand using your mouth. It is easy to detect even small amounts of suction. The amount needed to lift your hand would definitely be noticed.]

It is easy to picture the molecules of air hitting the bottom of your hand and lifting it up. It is also easy to determine that the air that is hitting your hand is bouncing off your hand and going downwards. You can hold your hand in such a way as to make the air bounce off your hand and hit your face. When you do this, your hand is pushed in a direction away from your face.

Whichever way your hand directs the air, the air pushes your hand in the opposite direction. The amount of force your hand feels depends on the amount of air that is pushed in the opposite direction. If you hold something larger than your hand out the window of the car, you will feel a larger force, since more air is being moved. The amount of force your hand feels also depends on the speed of the car. The faster the wind, the larger the force. Lastly, the density of the air affects the force you feel. If you put your hand in the water when you travel by boat, you feel a much larger force, even if the boat is going much slower than the car.

Lift and drag

Notice that you feel the largest lifting force on your hand when it is held at a 45 degree angle. This angle causes the wind to bounce off your hand straight down. But you also feel a force pushing your hand backwards, away from the direction of travel. This is because you are stopping the air molecules from travelling backward, and are making them go down instead. The backward force you feel is equal to the upward force.

The backward force (the force resisting forward motion) is called drag, and it is easy to see that we cannot get lift without drag. We cannot change the direction of the wind without feeling its resistance to change.

There is another type of drag that is important to understanding how the ball balances on the stream of air, or how airplanes fly, or how sailboats sail into the wind.

Let's look again at your hand moving through water. If your hand moves very slowly, it will not stir up the water very much behind it. If you move your hand more quickly, you will see little whirlpools form behind your hand. It takes force to cause all this water motion, and that force is felt as added resistance to the movement of your hand. If we could somehow streamline your hand to get the most movement of water in the direction we want, while causing the least stirring up of the water, we would have less drag for a given lift, and we would have a more efficient wing, or propeller, or sail.

Viscosity and drag

In water, we have to move very slowly to avoid causing whirlpools (also known as vortices). In air, we can move a little faster without stirring up the air in the same way (we can see this if we use smoke to make the vortices visible). In a jar of honey, we have to move very slowly. The difference is viscosity.

Viscosity is a property of fluids (like air or water). It is the ability of the fluid to resist changes in its shape that do not change its volume.

Viscosity is caused by interactions between the molecules of the fluid. It is the transfer of momentum from one part of the fluid to another part.

In a gas like air, viscosity is caused almost entirely by collisions between molecules. The faster the molecules are moving, the more effective is the transfer of momentum. In a hot gas, the molecules are moving faster than in a cold gas, so the viscosity is higher. A small part of viscosity in air is caused by attractive forces between molecules. These forces are much larger in water and honey, and play a bigger part there. These forces are called Van Der Waals forces. In air these forces are small enough to ignore when explaining lift and drag.

It is useful to compare the momentum forces in a fluid to the viscous forces. The ratio of the two is called the Reynolds number. It is defined as the density times the velocity times the length (width of your hand) all divided by the viscosity.

If the viscosity is low, like it is in air, or the speed is slow, then your hand does not transfer momemtum to very much air. Almost all of the momentum that is transfered goes into moving the air downwards, and very little goes into stirring it up. As the speed increases, or the viscosity increases, then more of the momentum is transfered to the air above, below, and behind your hand, and is wasted as extra drag.

Curved surfaces and the Coanda effect

If we limit the angle that the fluid has to turn as it passes the wing, then we can limit the rotation of the fluid. The fluid won't spin around as much if we don't kick it very much. If we put a cylinder in the water, we will see a lot of vortices created as it moves. If we shape it like a fish or a teardrop, where the trailing edge gradually tapers to a point, then the water does not have to turn as sharply, and so it does not spin as much, and smaller vortices are produced.

Now we can see why some wings are curved on the top. By gradually letting the air fill the empty space behind the wing, we limit the amount of spin we impart to the vortices. This limits the drag on the wing.

The tendency of a fluid to follow a curved surface is called the Coanda effect.

Notice what is happening in the following photograph of smoke pulses flowing over an airplane wing. The air slows down as it gets close to the wing. The Bernoulli principle says that this slower moving air will appear to the wing to have a higher pressure that faster moving air. What keeps this high pressure from pushing the wing down is the fact that it happens on the bottom of the wing as well, and is balanced.

Note also that the air does not speed up as it moves over the curved top of the wing, but it does slow down as it encounters the tilted bottom of the wing. We can measure the pressures on the top of the wing and on the bottom, and the difference is lift. We get the same value for lift whether we look at the mass of air moving downwards, or the pressure difference between the top of the wing and the bottom, because they are two different ways of looking at the same thing.

Notice that the viscosity of the fluid causes it to follow the shape of the wing. In a gas, the viscosity is the result of collisions between molecules. The molecules above the wing are constantly bumping into one another. As the wing sweeps away the molecules in front of it and pushes them downwards, it leaves an empty space behind it. The air above this empty space expands into it, due to the collisions of the molecules.

Picture the wing as having two springs attached to it, one pushing down on the top of the wing, and one pushing up on the bottom of the wing. If we move the wing down, we compress the bottom spring, and the top spring expands because we are no longer pushing on it as hard as we were before. The springs are the air molecules bouncing against the wing and each other. We are moving two masses of air in the downward direction. The air above the wing moves down as well as the air under the wing.

Before the wing moved the air, the air on the bottom was holding up the air on the top. In order to do this, there must have been a force pushing upwards. The wing moving through the air must not only accelerate the molecules in a downward direction, but it must overcome this upward force that is holding up the air above the wing. We can think of this upward force as helping to hold up the wing now instead of holding up the air above the wing. It thus adds to the lift on the wing. The air above the wing now falls into the empty space behind the wing. It also falls past the wing as the wing moves out of the way, and we can measure the amount of air that is moving down and see that it matches the lift on the wing as expected.

Back to why the ball balances...

We are now finally ready to see why the ball balances on the stream of air.

To balance, the ball must see a force that tends to center it on the air stream when it strays. From our discussion above, we would expect to see two things happen if this force exists. We would see air moving in the opposite direction of the ball's motion. We would also expect to see a higher pressure on the side of the ball opposite the air stream, and a lower pressure on the side facing the air stream.

Picture the air stream grazing the ball on the left side. The curve of the ball s fairly gentle, and causes the air to follow the curve. As the air follows the curve, it moves away from the stream of air. If the air is moving away from the stream of air, whatever caused it to move (the ball) must feel a force towards the stream of air.

This air on the left side is moving faster than the air on the right side (which isn't moving). As the air moves past the ball, it sweeps aside air molecules that were moving towards the ball, and would have hit the ball if they had not been moved aside. The pressure on that side of the ball is thus lower.

On the right side of the ball the air is not moving, so the pressure has not changed. The pressure on the left is lower than the pressure on the right, so the ball moves towards the stream of air.

No matter which direction the ball is deflected, it is attracted to the center of the air stream, and stays balanced.

The Bernoulli principle states that the pressure the ball sees on the side towards the moving air is less than on the side where the air is still. This is why we call the toy the "Bernoulli Ball'. Notice that what actually moves the ball is the recoil of lots more tiny air molecules on the right side of the ball than on the left. We could call the toy the "Newton Ball", but that lacks alliteration. Looking at it another way, we see air moving away from the stream of air as it follows the curve of the ball. So we could also call the toy the "Coanda Ball", although I prefer to think of the Coanda effect as the result of lift, not the cause.

Flux Density

2008-11-26T00:01:00.000-08:00

The emission from a blackbody,


			(1)

where is the Planck function, is the azimuthal angle, is the polar angle, and is a differential element of solid angle. The units are J Hz^-1 m^-2 s^-1.

The radio astronomical unit of flux is the Jansky.

Flux, Jansky, Planck Function

Laser

2008-11-25T00:01:00.000-08:00



				An acronym for "light amplification by stimulated emission of radiation." The first working laser was built by Maiman (1960), and made use of optical pumping of a ruby crystal from a flash lamp. The first continuous laser was produced by Nelson and Boyle (1962) using an arc lamp instead of a flash lamp. Natural lasing in the 10 m bands of CO₂ in the atmospheres of Mars and Venus was observed by Johnson et al. (1976), and subsequently studied in detail in the mesospheres and tropospheres of Mars (Mumma et al. 1981) and Venus (Deming and Mumma 1983). In these atmospheres, the population inversion is pumped by sunlight (Mumma et al. 1981, Deming and Mumma 1983, Gordiets and Panchenko 1983, Stepanova and Shved 1985, Dickinson and Bougher 1986, Mumma 1993). The distinction between lasing and masing in space sources is ambiguous in the infrared region of the spectrum, so that the claim that the "first natural laser" was observed at 169 m by Strelnitsky, Erickson, and Haas in Kuiper Airborne Observatory observations of the accretion disk surrounding the star MWC349 in Cygnus (Glanz 1995) is not very significant considering that 450 and 850 m "masing" had already been identified in the same source (Thum et al. 1994ab, Mumma 1996). Laser Cooling, Laser Trapping, Maser Bertolotti, M. Masers and Lasers: An Historical Approach. Bristol: A. Hilger, 1983. Bromberg, J. L. The Laser in America, 1950-1970. Cambridge, MA: MIT Press, 1991. Deming, D. and Mumma, M. J. "Modeling of the 10-m Natural Laser-Emission from the Mesospheres of Mars and Venus." Icarus 55, 356-368, 1983. Dickinson, R. E. and Bougher, S. W. J. Geophys. Res. 91, 70, 1986. Glanz, J. "First Light from a Space Laser." Science 269, 1336, 1995. Gordiets, B. F. and Panchenko, V. Ya. Cosmic Res. (U. S. A.) 21, 725, 1983. Hallmark, C. L. and Horn, D. Lasers, the Light Fantastic. Blue Ridge Summit, PA: Tab Books, 1979. Hecht, J. The Laser Guidebook, 2nd ed. New York: McGraw-Hill. Hecht, J. Laser Pioneers, rev. ed. Boston: Academic Press, 1992. Hecht, J. Understanding Lasers: An Entry-Level Guide. Indianapolis, IN: H.W. Sams, 1988. Hecht, J. and Teresi, D. Laser: Light of a Million Uses. New York: Dover, 1998. Johnson, M. et al. Astrophys. J. Let. 208, L145, 1976. Maiman, T. H. "Stimulated Optical Radiation in Ruby." Nature 187, 493-494, 1960. McAleese. The Laser Experimenter's Handbook. McComb. The Laser Cookbook: 88 Practical Projects. 1988. Mehta, P. C. and Rampal, V. V. Lasers and Holography.. 1993. Mumma, M. J. et al. Science 212, 45, 1981. Mumma, M. J. In Astrophysical Masers (Ed. A. W. Clegg and G. E. Nedoluha). Berlin: Springer-Verlag, pp. 455-467, 1993. Mumma, M. J. "Space Laser." Letter to Science 270, 717, 1996. Nelson, D. F. and Boyle, W. S. Appl. Opt. 1, 181, 1962. Shimoda, K. Introduction to Laser Physics: With 99 Figures, 2nd ed., corrected 2nd printing. Berlin: Springer-Verlag, 1991. Siegman, A. E. Lasers. Mill Valley, CA: University Science Books, 1986. Silfvast, W. T. Laser Fundamentals. Cambridge University Press, 1996. Stepanova, G. I. and Shved, G. M. Sov. Astron. Let. 11, 390, 1985. Talbot, J. "Optical Laser." `http://www.achilles.net/~jtalbot/history/ruby.html`. Thum, C.; Matthews, H. E.; Martinpintado, J.; Serabyn, E.; Planesas, P.; et al. "A Submillimeter Recombination Line Maser in MWC-349." Astron. Astrophys. 283, 582-592, 1994a. Thum, C.; Matthews, H. E.; Harris, A. I.; Tacconi, L. J.; Schuster, K. F. et al. "Detection of H21 Maser Emission at 662 GHz in MWC-349." Astron. Astrophys. Let. 288, L125-L128, 1994b. Townes, C. H. How the Laser Happened: Adventures of a Scientist. Oxford, England: Oxford University Press, 1999. van Hecke, G. and Karukstis, K. K. A Guide to Lasers in Chemistry. Jones and Bartlett, 1998. Weisstein, E. W. "Books about Lasers." http://www.ericweisstein.com/encyclopedias/books/Lasers.html. Zare, R. N.; Spencer, B. H.; Jacobson, M. P. Laser Experiments for Beginners. New York: Springer-Verlag, 1995. © 1996-2007 Eric W. Weisstein

Celestial Mechanics

2008-11-24T00:01:00.000-08:00

Celestial Mechanics

Apoapsis

Argument of Pericenter

Celestial to Equatorial C...

Eccentricity Perturbations

Elliptical Orbit

Equation of Center

Escape Velocity

Euler's Two-Point Orbit D...

F- and G-Functions

f- and g-Series

Gauss-Encke-Merton Method

Gauss's Principle of Leas...

Gaussian Gravitational Co...

Gaussian Vectors

Geostationary Orbit

Geostationary Transfer Or...

Herrick and Liu Method

High Earth Orbit

Hodograph

Hohmann Transfer Orbit

Lagrange's Planetary Equa...

Laplace-Runge-Lenz Vector

Libration in Latitude

Libration in Longitude

Line of Apsides

Line of Nodes

Longitude at Epoch

Longitude of Pericenter

Longitude of the Ascendin...

Mean Longitude at Epoch

Mean Motion

Molniya Orbit

n-Body Problem

Nonsynchronous Rotation

Nutation

Obliquity

Obliquity-Induced Precess...

Orbital Reference System

Orbital Resonance

Orbital Stability

Osculating Orbital Elemen...

Precession of the Equinox...

Reduced Mass

Reference Plane

Relativistic Precession

Relativistic Two-Body Pro...

Restricted Three-Body Pro...

Specific Angular Momentum

Stumpff Function

Sun-Synchronous Orbit

Three-Body Problem

Time of Pericenter Passage

Hubble 15 Years of Discovery | PDF | 30.6 MB -=RS.COM=-

2008-11-23T07:52:00.000-08:00

From the reviews:

"This is the story of a journey through space and time revealed by a telescope called Hubble. … published as part of the European Space Agency’s 15th anniversary celebration, presents the exquisite color images for which the telescope has become famous. … Hubble has pushed the limit of our knowledge far beyond anything possible before its launch. … Super-sharp images, such as those found here, have enabled astronomers to gain entirely new insights into the workings of a huge range of different astronomical objects."

Product Description:

Hubble: 15 Years of Discovery forms a key element of the European Space Agency's 15th anniversary celebration activities for the 1990 launch of the NASA/ESA Hubble Space Telescope.

As an observatory in space, Hubble is one of the most successful scientific projects of all time, both in terms of scientific output and its immediate public appeal.

Hubble continues to have an enormous impact by exploiting a unique scientific niche where no other instruments can compete. It consistently delivers super-sharp images and clean, uncontaminated spectra over the entire near-infrared and ultraviolet regions of the electromagnetic spectrum. This has opened up new scientific territory and resulted in many paradigm-breaking discoveries.

DOWNLOAD

Check my Blog Daily for New Posts. Not all posts make it to front page..;-)

Magnetic Monopoles in Spin Ice?

2008-11-22T00:01:00.000-08:00

When I wrote about the Kondo effect a while back, I mentioned that there were many fundamental physics that came out of the field of condensed matter, one of them being the inspiration for the Higgs mechanism.

Now comes another theoretical discovery from condensed matter that could shed light on the on-going search for the magnetic monopoles. It appears that such a thing could be found in, of all places, a magnetic material called the spin ice. Again, just like the fractional quantum hall system, the dimensional effect in a strongly-correlated electron system can produce such a rich set of phenomena, it is just a zoo of basic, fundamental physics waiting to be discovered.

This paper has been published in Nature[1]. Also don't miss the News and Views review of this paper by Oleg Tchernyshyov in the same issue of Nature.

One environment in which monopoles might pop up is crystalline solids. In a crystal at a low temperature, excitations above the ground state often behave like elementary particles: they carry a quantized amount of energy, momentum, electric charge and spin. In their theoretical study, Castelnovo et al. find the first instance of such an excitation with a non-zero magnetic charge. Under certain conditions, these magnets behave as a gas of independent magnetic poles. There is even a phase transition at which a thin vapour of these monopoles condenses into a dense liquid.

Moral of the story: you CAN study some of the most fundamental aspect of our world in Condensed matter physics. It is as fundamental as any. Just because the field has a direct application to the study of the properties of materials doesn't make it any less fundamental.

Zz.

[1] C. Castelnovo et al, Nature v.451, p.42 (2008).

A simple laser communicator.

2008-11-21T07:06:00.000-08:00

How would you like to talk over a laser beam? In about 15 minutes you can set up your own laser communication system, using cheap laser pen pointers and a few parts from Radio Shack.

For the transmitter you will need:

A laser pen pointer. You can get one for $10 from our catalog.
A battery holder that holds the same number of batteries as the laser pointer (often 3 cells). The batteries can be any size, but they must be the same voltage as the laser batteries. You may need to get one that holds two cells, and another that holds one cell, and wire them together in series. Radio Shack has a decent selection.
A transistor radio. Later we will use a microphone and an amplifier (Radio Shack #33-1067 and #277-1008), but at first we will send your favorite radio station over the laser beam.
An earphone jack that will fit your transistor radio (Radio Shack #42-2434).
A transformer of the type known as an audio output transformer. It consists of an 8 ohm coil and a 1000 ohm coil. The one I used is the Radio Shack #273-1380. We now carry them in our catalog.
Some clip leads (wires with alligator clips on the ends) to put it all together. At least one of the clip leads should be the type with a long slender point (Radio Shack #278-016, #270-372, or #270-334), to connect to the inside of the laser pointer. You can substitute regular wire and solder if you like, but the clip leads are fast and simple. Radio Shack has a wide selection of clip leads (such as ##270-378).
A two-lead bicolor light emitting diode, to protect the laser from high voltage spikes.

For the receiver you will need:

Make a solar powered marshmallow roaster

2008-11-20T08:24:00.000-08:00

In this section we will describe how to make a marshmallow roaster, powered by the sun. It can be made from readily available materials, and while it is probably a little safer than the traditional method of roasting marshmallows (over a campfire), it can still start fires, and should be used only by those you would trust with a box of matches.

Like most of the projects in this book, it is not just a fun toy, but a toy that teaches important scientific principles.

What we will need:

A page magnifier. More technically known as a Fresnel lens (pronounced freh-nell), this is a piece of plastic almost the size of a notebook page (7 inces by 10 inches) that is used to magnify a page of a book to make it easier to read. These are available in drug and stationery stores, and also in our catalog.
A small cardboard box. The actual size will depend on the focal length of the magnifier. The magnifier we use focuses the sun to a small bright dot at a point 10 and a half inches from the lens. This means that a box 10 inches on a side would be perfect. The actual dimensions are not critical, as we will discuss ways to adjust for large or small boxes.
A package of bamboo skewers to hold the marshmallows. You can also use coat-hanger wire, or long fondue forks.
Some aluminum foil.
Some glue.
Some tape.

Click on the photo for larger picture

The first step is to cut a hole in the box just 1/4 inch smaller on each edge than the Fresnel lens.

Click on the photo for larger picture

Then we tape the lens to the inside of the box. The lens has a smooth side and a grooved side. The grooved side should be facing out, away from the inside of the box.

Click on the photo for larger picture

Next, we glue aluminum foil to the inside of the box, on all sides except the side that has the lens. This is to make sure that if the box is accidentally left in the sun, the lens will reflect off of the shiny aluminum, and not burn a hole in the cardboard.

The foil should be shiny side out, and it does not matter if it is wrinkled.

Click on the photo for larger picture

on the side of the box opposite the lens, we now cut a square hole, about twice the size of a marshmallow.

On either side of the hole, we cut small triangular tabs to hold the skewer. These tabs are bent outward, and the skewer rests on them.

Click on the photo for larger picture

The photo above shows the results of a somewhat overzealous approach to marshmallow roasting. While the outside maybe quite overdone, you can see from the drooping position of the result on the skewer, the inside is a warm creamy delight.

The bright sunlight, concentrated on the highly reflective white marshmallow, is difficult to look at. Welding goggles, an inexpensive item at most hardware stores, adds an extra level of excitement and awe to the participants. Very dark glasses (or two pair of dark glasses), or solar eclipse viewing glasses, also work well.

A light coating of chocolate syrup or cocoa powder helps the marshmallow absorb the sunlight instead of reflecting it. This speeds up the roasting process, and reduces the glare on the eyes. Some kids like their marshmallows "well done" and first burn a small black hole in the marshmallow by holding it at the exact focus of the lens, and then expand the black spot by moving away from the focus a little bit. The black spot absorbs the sunlight very well, and the marshmallow cooks quickly.

This roaster can also be used for vienna sausages, or bite-size pieces of hot dog.

How does it do that?

A flat plate of glass does not magnify. To magnify an image, the glass must have a curved shape, like a magnifying glass does. The name "lens" comes from the Latin word for the lentil, a seed which has a shape of a disk whose top and bottom surfaces curve outward.

But the Fresnel lens we used in the marshmallow roaster appears to be flat. This is because a special trick is used to make a flat magnifier.

Remember that we said a flat plate of glass does not magnify.

Inside a normal lens, we can draw many rectangular areas. These areas are glass, but since they have flat edges, they do not help the lens magnify. So they are not useful for the purpose of a magnifier, and simply add unnecessary weight and cost to the lens.

The second part of the drawing below shows what is left if we remove the useless parts, and only keep the parts of the lens that magnify.

Click on the drawing for larger picture

One side of our resulting "lens" is flat. But the other side has ridges with curved sides. These curved pieces of glass (or plastic in our Fresnel lens) bend the light in the same way as the original lens did. This discussion of how Fresnel lenses work is actually a simplification of what is really going on. We will explain in more detail later.

If you rub the Fresnel lens with your fingers, you can feel these ridges.

Absorption

Concentrating the sunlight is only half of what is going on in the roaster. The other half is what happens when the light hits the marshmallow.

The marshmallow is white. It reflects almost all of the light that hits it. Only a small fraction of the light is absorbed. When light is absorbed by a material, it is not lost. The energy from the light moves the molecules of the marshmallow. Moving molecules is what we feel as heat.

In order to heat up the marshmallow, we had to use the very smallest dot of light from the lens, where all of the sunlight is concentrated into one tiny spot. The small fraction of the light that the marshmallow absorbs is now enough to heat up the marshmallow until it burns at that spot.

But now the burned part of the marshmallow is no longer white. It no longer reflects very much light. That is why it appears black. Black objects are those that absorb much more light than they reflect.

Now that the spot is absorbing most of the sunlight, it gets hot very quickly. If we don't move the marshmallow, it will catch fire.

We move the marshmallow closer to the lens, so the circle of light from the lens is bigger, and thus less concentrated. It is still concentrated enough to roast the black spot on the marshmallow, and make it bigger.

By coating the marshmallow with a dark substance, like chocolate syrup or cocoa, we can speed up the heating of the marshmallow.

More about Fresnel lenses

Our discussion about how Fresnel lenses work, we gave the standard textbook explanation, which explains the concept, but misses some details that are important if you want to do real work with the lens.

In the simplified example, we simply moved the curved pieces down to lie flat. But a curve that is designed to focus light onto a point depends on the middle of the lens being farther away from the focal point than the edge. If we simply moved the pieces down, they would not focus the light to a point. The edges would focus the light to the same point as before, but as we move to the center of the lens, the focal point moves farther away, by the same amount that we moved the pieces down.

Real Fresnel lenses compensate for this. The curves are made to keep the focus at the same point, regardless of how close to the center of the lens a light ray is.

Fresnel lenses are usually flat on one side. The corrections made to keep the focus at a point only work from one direction. The lenses are most commonly made to focus light in such a way that the grooved side must face the sun, and the flat side must face the focal point. If the lens is reversed, it will not focus to a sharp point. The edges will focus too close, and the center will focus too far away. This is why we said to make sure the grooved side of the lens faced outside the box (towards the sun).

Fun with a big lens

The photo below shows a Fresnel lens boiling water in a frying pan on my driveway. The pan is set well above the focal length, so it won't melt. The size of the spot of light is just a bit smaller than the frying pan, about 6 inches across.

Click on the drawing for larger picture

The frame is made of 1 inch by 4 inch lumber, supported by 2 inch by four inch common studs. The lens itself is 40 inches across, and 30 inches high. It came from a 40 inch projection television set.

Below is a photo of four U.S. pennies that were placed at the actual focus, a spot of light about the size of the hole in the penny (1/4 inch across).

Click on the drawing for larger picture

Three of the pennies were of the old copper alloy type (pre-1982). They contain 95% copper and 5% zinc. The penny on top was the new copper plated zinc type. It is 97.6% zinc (all in the center) and 2.4% copper (all in the plating). Zinc has a lower melting point than brass. Much of the zinc actually burned away, leaving the pitted surface you can see in the photo. The copper plating melted and dissolved into the zinc, making the bright gold colored brass lump that joins the other pennies together. I then moved the focus to the bottom penny, and melted the hole in it. The entire procedure took only three or four seconds.

Faraday cage

2008-11-19T20:48:00.000-08:00

From Wikipedia, the free encyclopedia

Entrance to a Faraday room

A Faraday cage or Faraday shield is an enclosure formed by conducting material, or by a mesh of such material. Such an enclosure blocks out external static electrical fields. Faraday cages are named after physicist Michael Faraday, who built one in 1836.^{[citation needed]}

An external static electrical field will cause the electrical charges within the conducting material to redistribute themselves so as to cancel the field's effects in the cage's interior. This effect is used, for example, to protect electronic equipment from lightning strikes and other electrostatic discharges.

To a large degree, Faraday cages also shield the interior from external electromagnetic radiation if the conductor is thick enough and any holes are significantly smaller than the radiation's wavelength. For example, certain computer forensic test procedures of electronic components or systems that require an environment devoid of electromagnetic interference may be conducted within a so-called screen room. These screen rooms are essentially labs or work areas that are completely enclosed by one or more layers of fine metal mesh or perforated sheet metal. The metal layers are connected to earth ground to dissipate any electric currents generated from the external electromagnetic fields, and thus block a large amount of the electromagnetic interference. This application of Faraday cages is explained under electromagnetic shielding.

History

In 1836 Michael Faraday observed that the charge on a charged conductor resided only on its exterior and had no influence on anything enclosed within it. To demonstrate this fact he built a room coated with metal foil and allowed high-voltage discharges from an electrostatic generator to strike the outside of the room. He used an electroscope to show that there was no electric charge present on the inside of the room's walls.

The same effect was predicted earlier by Francesco Beccaria (1716–1781) at the University of Turin, a student of Benjamin Franklin, who stated that "all electricity goes up to the free surface of the bodies without diffusing in their interior substance." Later, the Belgian physicist Louis Melsens (1814–1886) applied the principle to lightning conductors. Another researcher of this concept was Gauss (Gaussian surfaces).

How a Faraday cage works

An external electrical field causes the charges to rearrange which cancels the field inside.

A Faraday cage is best understood as an approximation to an ideal hollow conductor. Externally applied electric fields produce forces on the charge carriers (usually electrons) within the conductor, generating a current that rearranges the charges. Once the charges have rearranged so as to cancel the applied field inside, the current stops.

If a charge is placed inside an ungrounded Faraday cage the internal face of the cage will be charged (in the same manner described for an external charge) to prevent the existence of a field inside the body of the cage. However, this charging of the inner face would re-distribute the charges in the body of the cage. This charges the outer face of the cage with a charge equal in sign and magnitude to the one placed inside the cage. Since the internal charge and the inner face cancel each other out, the spread of charges on the outer face is not affected by the position of the internal charge inside the cage. So for all intents and purposes the cage will generate the same electric field it would generate if it was simply charged by the charge placed inside.

If the cage is grounded the excess charges will go to the ground instead of the outer face, so the inner face and the inner charge will cancel each other out and the rest of the cage would remain neutral. A Faraday cage is capable of completely stopping an attack using electromagnetism such as an EMP.^{[citation needed]}

The cage will block external electrical fields even if the cage contains some charges and an electric field in its interior. This is a consequence of the superposition principle and the fact that Maxwell equations are linear.

A Faraday cage will not shield its contents from static magnetic fields. However, rapidly-changing magnetic fields create electric fields in accordance with Maxwell's equations. The conductors cancel the electric fields and therefore the changing magnetic fields as well. The wall materials' thickness and skin depth set the frequency at which the cage suppresses electromagnetic fields. Static or slowly-changing magnetic fields penetrate the cage; rapidly-changing ones do not.

Real-world Faraday cages

MRI scanners are typically housed inside Faraday cages.
Mobile phones and radios may have no reception inside some elevators or similar structures.
Some traditional architectural materials act as Faraday shields in practice. These include plaster with metal lath, and rebar reinforced concrete. These affect the use of cordless phones and wireless networks inside buildings and houses.
Steel buildings and steel-clad buildings act as Faraday cages. Sheds are often steel-clad or made of steel.
The cooking chamber of the microwave oven itself is a partial Faraday cage enclosure which prevents the microwaves from escaping into the environment.
Electronic components that can be damaged by static charges, such as integrated circuits and computer cards, are shipped in Faraday cages consisting of special bags made of an electrically conductive plastic, called antistatic bags.
Coaxial cables are in fact data cables wrapped by a hollow, flexible conductor, effectively a Faraday cage.
RFID passport and credit card shielding sleeves are small, portable Faraday cages.
Some United States national security buildings which house a Sensitive Compartmented Information Facility are contained in Faraday cages, intended to act as a TEMPEST shield, and possibly also as a mitigation against electromagnetic pulse.
A teacher in the UK has come up with the idea to curb cheating (via text message using mobile phones) in examinations by lining every exam room with a Faraday-like cage.^[1]
Cars and aircraft function as Faraday cages when struck by lightning. The metal frame and outer skin of the vehicle cause the electrical charge to travel safely away from the occupants. This differs from a popular urban legend that claims that a car's tires cause the lightning strike to reach the ground. However, radio and cellular phone signals can still reach inside the vehicle since their wavelengths are significantly smaller than the windows and other openings in the vehicle's conductive frame.
- The role of a car as Faraday cage was demonstrated by Richard Hammond in an episode of the BBC television program Top Gear. At the Siemens High-Voltage Lab in Berlin, Germany, Hammond sat in a car that was being struck by simulated lightning of over 800,000 volts, and reported during the experiment that he felt nothing.^[2]
The Discovery Channel television show MythBusters used a Faraday cage made from a brass mesh to "cancel out" radio signals that might have interfered with the consistency of an experiment.
In scientific environments such as the National Radio Astronomy Observatory in Greenbank, West Virginia, Faraday cages are used to enclose computer equipment rooms that, despite being vital to the cause, interfere with experiments involving radio astronomy. The cages block the electromagnetic waves that skew data and could damage radio telescopes. Pulsed high-voltage experiments also use such Faraday cages to protect sensitive electronics from the experiments' electromagnetic pulses. In this context, the cages are often called "screen rooms".
Faraday cages have been built into wearable suits, allowing high-voltage workers to sit directly on power lines.^[3]
Wifi signals are often confined inside of a building, especially if the building has metal siding.
The internal metal lining of most consumer electronics, as well as the metal case of most personal computers, act as a Faraday cage to reduce interference to and from other devices.
A manufactured home with aluminum siding will contain many of the physical characteristics of a Faraday cage, with the exception of windows and flooring. This type of structure is noted for blocking NOAA Weather Radios from receiving accurate signals, especially if the receiver is improperly placed in the manufactured home.^[4]

The effectiveness of a Faraday cage or shield is dependent upon the wavelength of the electric or electromagnetic fields it is intended to shield. This explains why a microwave oven, for example, can perform such shielding from the observer peering through the metal mesh screened "window" at the front of the oven to watch the cooking process take place. The holes are sized such that the waves within the oven cannot pass through even though visible light which has a much shorter wavelength easily passes through the holes. This also explains how cell phones have improved in building performance using the higher frequencies (shorter wavelengths) of EMFs than the earlier predecessors, notwithstanding improved digital modulation algorithms in so called 3G handsets today and later standards forthcoming. Quality levels of shielding also depend upon the types of metals used in the cages as well as the thicknesses.

References

^ http://www.theregister.co.uk/2006/09/25/exam_cages/
^ "Top Gear Video: Richard Conducts". Retrieved on 2008-08-18.
^ glumbert - High Power Job
^ http://www.srh.noaa.gov/topics/attach/html/msb01-01.htm

Fun with High Voltage (part 4)

2008-11-19T08:35:00.000-08:00

A high voltage ion motor

This motor is very simple to build, and goes together in a few minutes. All you need is two pieces of wire, the small metal cap from the fuse we took apart in the previous project, and some cellophane tape.

The motor creates an ion wind that spins it around like a helicopter.

Click on the image for a larger picture

First, take one piece of wire (a straightened paper clip will do), and cut the end at an angle so it is sharp. Bend the other end into a rough loop or triangle, so the wire will stand up with the sharp point facing straight up. A little tape will help hold it onto the table, or a block of wood.

Click on the image for a larger picture

The armature (the part that spins) is made from the other piece of wire and the metal cap we saved when we took apart the fuse. Sharpen both ends of the wire by cutting the ends at a diagonal, like we did with the base wire. Bend the wire into an S shape. The pointed ends of the wire should point at 90 degrees from the center straight part of the wire.

Click on the image for a larger picture

Attach the metal cap to the center of the wire with tape. Place the cap onto the pointed end of the base wire, and bend the S shaped ends of the armature wire down, so it will balance easily on the sharp end of the base wire.

The armature should now spin freely if you tap it gently.

Connect a source of high voltage to the base wire using an alligator clip or a wire. The high voltage source can be the Van de Graaff generator, or just a couple square feet of aluminum foil pressed against the front of your television set, as we did in earlier projects.

As the high voltage is turned on, the armature will start to spin in the direction away from the sharp points. The Van de Graaff generator may need a good ground, or a person holding onto the ground wire. The television will give the motor a good kick every time it is turned on or off, and turning it on and off every second will get it spinning quite rapidly.

Fun with High Voltage (part 3)

2008-11-18T08:17:00.000-08:00

A simple homemade Van de Graaff generator

In the previous two projects, we stole high voltage from a television set to power our high voltage motors. In this project we will build a device that can generate 12,000 volts from an empty soda can and a rubber band.

The device is called a Van de Graaff generator. Science museums and research facilities have large versions that generate potentials in the hundreds of thousands of volts. Ours is more modest, but is still capable of drawing 1/2 inch sparks from the soda can to my finger. The spark is harmless, and similar to the jolt you get from a doorknob after scuffing your feet on the carpet.

To build the toy, you need:

An empty soda can
A small nail
A rubber band, 1/4 inch by 3 or 4 inches
A 5x20 millimeter GMA-Type electrical fuse (such as Radio Shack #270-1062)
A small DC motor (such as Radio Shack #273-223)
A battery clip (Radio Shack #270-324)
A battery holder (Radio Shack #270-382)
A styrofoam cup (a paper cup will also work)
A hot glue gun (or regular glue if you don't mind waiting)
Two 6 inch long stranded electrical wires (such as from an extension cord)
Two pieces of 3/4 inch PVC plumbing pipe, each about 2 or 3 inches long
One 3/4 inch PVC coupler
One 3/4 inch PVC T connector
Some electrical tape
A block of wood

That sounds like a lot of stuff, but take a look at the step-by-step photos below, and you will find that the whole project can easily be put together in an evening, once all the parts have been collected.

We'll start at the bottom, and work our way up.

Click on the image for a larger picture

The first thing to do is to cut a 2 to 3 inch long piece of 3/4 inch PVC pipe, and glue that to the wooden base. This piece will hold the generator up, and allow us to remove it to more easily replace the rubber band, or make adjustments.

The PVC "T" connector will hold the small motor. The motor fits too loosely by itself, so we wrap paper or tape around it to make a snug fit. The shaft of the motor can be left bare, but the generator will work a little better if it is made fatter by wrapping tape around it, or (better) putting a plastic rod with a hole in the center onto the shaft to act as a pulley for the rubber band.

Next, we drill a small hole in the side of the PVC "T" connector, just under the makeshift pulley on the motor. This hole will be used to hold the lower "brush", which is simply a bit of stranded wire frayed at the end, that is almost touching the rubber band on the pulley.

As the photo shows, the stranded wire is held in place with some electrical tape, or some other tape or glue.

The rubber band is now placed on the pulley, and allowed to hang out the top of the "T" connector.

Click on the image for a larger picture

Next, cut another 3 or 4 inch piece of 3/4 inch PVC plumbing pipe. This will go into the top of the "T" connector, with the rubber band going up through it. Use the small nail to hold the rubber band in place, as in the photo below. The length of the PVC pipe should be just enough to fit the rubber band. The rubber band should not be stretched too tightly, since the resulting friction would prevent the motor from turning properly, and increase wear on the parts.

Click on the image for a larger picture

Cut the styrofoam cup about an inch from the bottom, and carefully cut a 3/4 inch diameter hole in the center of the bottom of the cup. This hole should fit snugly onto the 3/4 inch PVC pipe.

Click on the image for a larger picture

Now drill three holes near the top of the PVC union coupling. Two of these holes need to be diametrically opposite one another, since they will hold the small nail which will act as an axle for the rubber band. The third hole is between the other two, and it will hold the top "brush", which, like the bottom brush, will almost touch the rubber band.

The top brush is taped to the PVC union coupler, and the coupler is placed on the 3/4 inch pipe, above the styrofoam cup collar. The rubber band is threaded through the coupler, and held in place with the small nail, as before.

Bare the top brush (so it has no insulation) and twist it to keep the individual wires from coming apart. You can solder the free end if you like, but it is not necessary.

The free end of the top brush will be curled up inside the empty soda can when we are done, and thus electrically connect the soda can to the top brush.

Click on the image for a larger picture

We need a small glass tube to act as both a low-friction top pulley, and as a "triboelectric" complement to the rubber band, to generate static electricity by rubbing. Glass is one of the best materials to rub against rubber to create electricity.

We get the tube by taking apart a small electrical fuse. The metal ends of the fuse come off easily if heated with a soldering iron or a match. The solder inside them drips out when they come off, so be careful. The glass, the metal cap, and the molten solder are all quite hot, and will blister the skin if you touch them before they cool.
Save the metal caps -- we will use them in a future project!

Click on the image for a larger picture

The resulting glass tube has nice straight, even edges, which are "fire polished" for you, so there is no sharp glass, and no uneven edges to catch on the PVC and break the glass.

The next step is a little tricky. The small nail is placed through one of the two holes in the PVC union coupler, and the small glass tube is placed on the nail. Then the rubber band is placed on the glass tube, and the nail is then placed in the second hole. The rubber band is on the glass tube, which is free to rotate around the nail.

Click on the image for a larger picture

Now we glue the styrofoam collar in place on the PVC pipe. I like to use a hot glue gun for this, since the glue can be laid on thickly to stabilize the collar, and it sets quickly and does not dissolve the styrofoam.

Click on the image for a larger picture

At this point we are ready for the empty soda can. Aluminum pop-top cans are good for high voltage because they have nice rounded edges, which minimizes "corona discharge".

With a sharp knife, carefully cut out the top of the soda can. Leave the nice crimped edge, and cut close to the side of the can so as to leave very little in the way of sharp edges. You can smooth the cut edge by "stirring" the can with a metal tool like a screwdriver, pressing outward as you stir, to flatten the sharp edge.

Tuck the free end of the top brush wire into the can, and invert the can over the top of the device, until it rests snugly on the styrofoam collar.

Click on the image for a larger picture

The last step is to attach the batteries. I like to solder a battery clip to the motor terminals, and then clip this onto either a nine-volt battery, or a battery holder for two AA size batteries. The nine-volt battery works, but it runs the motor too fast, making a lot of noise, and risking breakage of the glass tube. It does, however, make a slightly higher voltage, until the device breaks.

Click on the image for a larger picture

To use the Van de Graaff generator, simply clip the battery to the battery clip. If the brushes are very close to the ends of the rubber band, but not touching, you should be able to feel a spark from the soda can if you bring your finger close enough. It helps to hold onto the free end of the bottom brush with the other hand while doing this.

Click on the image for a larger picture

To use our generator to power the Franklin's Bells we built in the previous section of the book, clip the bottom brush wire to one "bell", and attach a wire to the top of the generator, connecting it to the other "bell".

The pop-top clapper of the Franklin's Bells should start jumping between the soda cans. It may need a little push to get started.

Click on the image for a larger picture

How does it do that?

You may have at one time rubbed a balloon on your hair, and then made the balloon stick to the wall. If you have never done this, try it!

The Van de Graaff generator uses this trick and two others to generate the high voltage needed to make a spark.

The first trick

When the balloon made contact with your hair, the molecules of the rubber touched the molecules of the hair. When they touched, the molecules of the rubber attract electrons from the molecules of the hair.

The you take the balloon away from your hair, some of those electrons stay with the balloon, giving it a negative charge.

The extra electrons on the balloon repel the electrons in the wall, pushing them back from the surface. The surface of the wall is left with a positive charge, since there are fewer electrons than when it was neutral.

The positive wall attracts the negative balloon with enough force to keep it stuck to the wall.

If you collected a bunch of different materials and touched them to one another, you could find out which ones were left negatively charged, and which were left positively charged.

You could then take these pairs of objects, and put them in order in a list, from the most positive to the most negative. Such a list is called a Triboelectric Series. The prefix Tribo- means "to rub".

The Triboelectric series

Most positive
(items at this end lose electrons)

asbestos
rabbit fur
glass
hair
nylon
wool
silk
paper
cotton
hard rubber
synthetic rubber
polyester
styrofoam
orlon
saran
polyurethane
polyethylene
polypropylene
polyvinyl chloride (PVC pipe)
teflon
silicone rubber

Most negative
(items at this end steal electrons)

Our Van de Graaff generator uses a glass tube and a rubber band. The rubber band steals electrons from the glass tube, leaving the glass positively charged, and the rubber band negatively charged.

The second trick

The triboelectric charging is the first trick. The second trick involves the wire brushes.

When a metal object is brought near a charged object, something quite interesting happens. The charged object causes the electrons in the metal to move. If the object is charged negatively, it pushes the electrons away. If it is charged positively, it pulls the electrons towards it.

Electrons are all negatively charged. Because like charges repel, and electrons are all the same charge, electrons will always try to get as far away from other electrons as possible.

If the metal object has a sharp point on it, the electrons on the point are pushed by all of the other electrons in the rest of the object. So on a point, there are a lot of electrons pushing from the metal, but no electrons pushing from the air.

If there are enough extra electrons on the metal, they can push some electrons off the point and into the air. The electrons land on the air molecules, making them negatively charged. The negatively charged air is repelled from the negatively charged metal, and a small wind of charged air blows away from the metal. This is called "corona discharge", because the dim light it gives off looks like a crown.

The same thing happens in reverse if the metal has too few electrons (if it is positively charged). At the point, all of the positive charges in the metal pull all the electrons from the point, leaving it very highly charged.

The air molecules that hit the metal point lose their electrons to the strong pull from the positive tip of the sharp point. The air molecules are now positive, and are repelled from the positive metal.

The third trick

There is one more trick the Van de Graaff generator uses. After we understand the third trick, we will put all of the tricks together to see how the generator works.

We said earlier that all electrons have the same charge, and so they all try to get as far from one another as possible. The third trick uses the soda can to take advantage of this feature of the electrons in an interesting way.

If we give the soda can a charge of electrons, they will all try to get as far away from one another as possible. This has the effect of making all the electrons crowd to the outside of the can. Any electron on the inside of the can will feel the push from all the other electrons, and will move. But the electrons on the outside feel the push from the can, but they do not feel any push from the air around the can, which is not charged.

This means that we can put electrons on the inside of the can, and they will be pulled away to the outside.

We can keep adding as many electrons as we like to the inside of the can, and they will always be pulled to the outside.

Putting all three tricks together

So now let's look at the Van de Graaff generator with our three tricks in mind.

The motor moves the rubber band around and around. The rubber band loops over the glass tube and steals the electrons from the glass.

The rubber band is much bigger than the glass tube. The electrons stolen from the glass are distributed across the whole rubber band.

The glass, on the other hand, is small. The negative charges that are spead out over the rubber band are weak, compared to the positive charges that are all concentrated on the little glass tube.

The strong positive charge on the glass attracts the electrons in the wire on the top brush. These electrons spray from the sharp points in the brush, and charge the air. The air is repelled from the wire, and attracted to the glass.

But the charged air can't get to the glass, because the rubber band is in the way. The charged air molecules hit the rubber, and transfer the electrons to it.

The rubber band travels down to the bottom brush. The electrons in the rubber push on the electrons in the wire of the bottom brush. The electrons are pushed out of the wire, and into whatever large object we have attached to the end of the wire, such as the earth, or a person.

The sharp points of the bottom brush are now positive, and they pull the electrons off of any air molecules that touch them. These positively charged air molecules are repelled by the positively charged wire, and attracted to the electrons on the rubber band. When they hit the rubber, they get their electrons back, and the rubber and the air both lose their charge.

The rubber band is now ready to go back up and steal more electrons from the glass tube.

The top brush is connected to the inside of the soda can. It is positively charged, and so attracts electrons from the can. The positive charges in the can move away from one another (they are the same charge, so they repel, just like electrons). The positive charges collect on the outside of the can, leaving the neutral atoms of the can on the inside, where they are always ready to donate more electrons.

The effect is to transfer electrons from the soda can into the ground, using the rubber band like a conveyor belt. It doesn't take very long for the soda can to lose so many electrons that it becomes 12,000 volts more positive than the ground.

When the can gets very positive, it eventually has enough charge to steal electrons from the air molecules that hit the can. This happens most at any sharp points on the can. If the can were a perfect sphere, it would be able to reach a higher voltage, since there would be no places where the charge was more concentrated than anywhere else.

If the sphere were larger, an even higher voltage could be reached before it started stealing electrons from the air, because a larger sphere is not as "sharp" as a smaller one.

The places on our soda can where the curves are the sharpest are where the charge accumulates the most, and where the electrons are stolen from the air.

Air ionizes in an electric field of about 25,000 volts per inch. Ionized air conducts electricity like a wire does. You can see the ionized air conducting electricity, because it gets so hot it emits light. It is what we call a spark.

Since our generator can draw sparks that are about a half inch long, we know we are generating about 12,500 volts.

Troubleshooting

If you aren't detecting any high voltage (no sparks, doesn't attract hair or paper) then you might try some of these suggestions.

Try a different type of rubber band. Some are slightly conductive, which at 12,000 volts means conductive enough to leak all the current you have so carefully built up. Have a supply of many different types of rubber band to try.
Make sure everything is very clean. Dirt and grease can be slightly conductive, and that will be enough to make the device fail.
Make sure the top brush is touching the metal of the can. Some cans have a plastic coating inside. Scrape it off (or burn it off) to make a better connection.
Make sure there are no sharp points extending outside the can. It is OK to have sharp points pointing inside the can, from the cut part of the top. Sharp points cause corona losses.
Make sure the brushes are not touching the rubber band. This will put a coating of copper on the rubber, and make it conductive.
Make sure you have a good ground connection.
Make sure the motor is spinning fast.

Check our Message Board for more ideas, and be sure to search for "VDG" and "rubber band" to get all of the messages. Since people can't spell Van de Graaff, you may want to try various spellings.

Some fun with the Van de Graaf generator

One of the fun things to do with a Van de Graaff generator is to show how like charges repel.

We take a paper napkin, and cut thin strips of the lightweight paper. We then tape the ends of the paper together at one end, and tape that end onto the Van de Graaf generator.

The effect will look somewhat like long hair cascading down the soda can.

Now turn the Van de Graaff generator on. The thin strips of paper all get the same charge, and start to repel from one another. The effect is "hair raising". The strips start to stand out straight from the can, like the hair on the back of a scared cat.

Click on the image for an animated movie

Fun with High Voltage (part 2)

2008-11-17T07:54:00.000-08:00

A rotary high voltage motor

Click on the photo to see an animated picture

At one company I once worked for, we had a contest for "The most creative use of office supplies". This toy would clearly be a contender. It is based on a wonderful design by Bill Beaty, but this version uses only things found around the typical office coffee room.

Using the safe high voltage power we get by placing a sheet of aluminum foil on the face of a television or computer CRT screen, it spins a styrofoam cup around at a respectable speed.

To build the toy, you need:

Two empty soda cans.
A styrofoam cup (a paper cup will also work).
A ball-point pen (the simple non-clicking type).
A couple square feet of aluminum foil.
Two paper clips
A hot glue gun (or regular glue if you don't mind waiting).
Cellophane tape.
Two wires (alligator test leads work great).

We start by spreading glue over the ouside of the styrofoam cup. Put just a thin layer on, so it dries quickly. Before it dries, cover the cup with aluminum foil. Press the foil flat against the cup, so any wrinkles are pressed down.

With a sharp knife, neatly cut a half inch strip out of the foil on both sides, so you have two patches of foil, one on each side of the cup, that do not touch one another.

The cup is going to be spinning upside down on the point of a ball-point pen. To keep the cup centered on the pen-point, and to provide a low friction bearing, we need to glue something hard to the center of the bottom of the cup, something that has a little dimple in it to sit on the pen-point.

I chose to sacrifice the end of another ball-point pen.

The photo shows the end of the pen, cut off with a sharp knife. The side of the cut end that is facing down has a little dimple that is perfect as a place to accept the point of the other ball-point pen.

The end of the pen is glued in the exact center of the bottom of the cup, as shown below. Note the little dimple.

Next we make the stand for the motor. Start with a paper plate, and glue the bottom of a ball point pen to the exact center of the plate, so the point stands straight up.

Glue the two soda cans upside-down onto the plate, leaving enough room between them for the strofoam cup to rotate easily without touching either can. There should ba about a half of an inch gap between the cup and either can.

Straighten two of the bends of a paper clip (leaving one end bent as in the photo below) and tape them to the cans as shown. Bend the wires into an S shape, leaving enough room to place the cup on top of the pen.

Now put the cup upside-down onto the pen-point. Make sure the dimple fits onto the pen-point. The wires should be about a half inch away from the cup, with the point being closest to the cup. Nothing should be touching the cup except the point of the pen.

Now connect a wire from the can on the right to a large sheet of aluminum foil pressed against the screen of a TV (or a computer with a CRT screen).

Connect another wire to the left can, and connect the free end to a good ground connection, such as a cold water pipe, or the metal frame of a computer. In a pinch, you can just hold onto the free end, since your body is a good enough ground for this little motor.

When you turn on the television, the foil will pick up a high voltage, and the little motor will start spinning. As it slows down, turn the television off, and the motor will get another kick, and spin faster. You can keep this up as long as you feel like turning the TV on and off.

Fun with High Voltage (part 1)

2008-11-16T07:42:00.000-08:00

A high voltage motor in 5 minutes

This toy is so simple to build, it goes together in 5 minutes from a few things found around the house.

The toy is a high voltage motor that acts like a bell, with a clapper that bangs furiously from one can to the other and back again, sometimes several times per second.

Occasionaly, the big blue spark snaps between the cans, to add interest to frenetic activity.

To build the toy, you need:

Two empty soda cans.
A plastic rod such as a ball-point pen.
5 inches of sewing thread.
A couple square feet of aluminum foil.
Cellophane tape.
Two wires (alligator test leads work great).

The photo may be all you need to get the toy working.

Remove the pull-tops from both cans, and discard one of the pull-tops.

Tie the thread to the other pull-top. Tape the other end of the thread to the center of the plastic rod.

Place the two cans side-by-side two or three inches apart.

Place the plastic rod on top of the two cans, so the pull-tab dangles freely about an inch from the table.

Tape the bare end of one wire to the left can. This is the ground wire, and the free end should be connected to an electrical ground, such as a cold water pipe, or the metal frame of a computer. If a good electrical ground is not convenient, you can just hold onto the free end, since your body is a good enough ground for this device.

Tape the other wire to the can on the right. It's free end will be connected to a source of high voltage. This is easier than it sounds, since a safe source of high voltage is right in front of you when you watch television or use a computer with a CRT monitor.

In the photo above, you can see that the toy is sitting on top of a television. About two square feet of aluminum foil is pressed onto the face of the TV screen. It sticks there because the TV screen is highly charged with electricity. The free end of the right can's wire is attached to the aluminum foil.

You start the toy by turning on the TV. The pull-top gets pulled to one can, but when it hits it, it gets pulled to the other can, and then repeats.

How does it do that?

Inside a television, a high voltage is used to send electrons to the screen at high speed, to create the picture. By placing a large conductor on the front of the screen, we can make a capacitor, to tap into some of that high voltage and put it to use outside of the television. The voltage is high, but the current is very small, so that touching the foil or the toy is no more harmful than touching a doorknob after scuffing your feet on the carpet.

The can on the right is connected to the high voltage. The can on the left is connected to the ground, which can absorb all of the voltage we can send it, and still be ready for more.

The pull-tab and the can on the left starts out without any electrical charge. We say they are at "ground potential".

The can on the right is charged with a lot of free electrons, from the foil on the TV screen. These electrons repel the electrons in the pull-tab, and attract the positive nuclei in the pull-tab.

The electrons in the pull-tab move to the side farthest from the high voltage can on the right. This leaves the right side of the pull-tab more positive than the left side. The positive side of the pull-tab is attracted to the highly negative can on the right, and the pull-tab jumps over to touch the can on the right.

Once it touches the can, the electrons from the can rush onto the pull-tab, until it has the same high voltage charge as the can it is touching. The pull-tab and the can now have the same charge, and like charges repel. The charged pull-tab is now repelled by the can on the right, and moves to the left.

The electrons in the can on the left are repelled by the pull-tab, and they move to the left side of the can, leaving the right side somewhat positive. This positive side attracts the negatively charged pull-tab, and draws it up to touch the can.

Now the excess electrons on the pull-tab move onto the left can, and into the ground. The pull-tab is now at ground potential again. It swings back towards the can on the right, and the whole process starts over again.

Fundamental Forces

2008-11-15T09:24:00.000-08:00

The basic forces in nature
Click on image for full size (32K GIF)
Image courtesy of Contemporary Physics Education Project

There are four forces in the Universe: strong, weak, electromagnetic and gravity. You may be most familiar with gravity. Gravity is what holds you and your dog to the surface of this planet!

The other forces may be less familiar to you, but they are very important! For example, the electromagnetic force holds electrons in orbit around the nucleus of an atom. We need these atom parts to be held together because we're made of atoms!

FUNDAMENTAL FORCES
Interaction	Relative Strength	Range	Mediating Particle
Strong	1	Short	Gluon
Electromagnetic	0.0073	Long	Photon
Weak	10^-9	Very Short	W,Z
Gravitational	10^-38	Long	Graviton

Table courtesy of University of Guelph, Guelph, Ontario (Cananda)

Gravity Definition Page

2008-11-14T09:22:00.000-08:00

An artist's rendering of some of the forces of the universe. The apple falling is of course from the story of Isaac Newton discovering the law of gravity as an apple fell from a tree he was sitting underneath.
Click on image for full size (42K GIF)
Windows Original

Gravity is one of the universal forces of nature. It is an attractive force between all things. The gravitational force between two objects depends on their masses, which is why we can really only see gravity in action when at least one of the objects is very large (like the Earth).

Isaac Newton was the first scientist to define gravity using math. There will be a larger pull due to gravity if the objects' masses are larger and if the objects are closer together.

One thing we can do with Newton’s law is calculate an escape velocity for the Earth. That is the speed something has to go for it to escape the gravity of Earth. This number for Earth is about 11 km/s. This means that if you could throw a baseball at 11 km/s, it would never come down!

Fundamental Physical Constants

2008-11-13T09:19:00.000-08:00

Planck constant h: 6.6260755·10^-34 J·s; h / (2 ) = 1.05457266·10^-34 J·s
Boltzmann constant k_B: 1.380658·10^-23 J/K ( = 8.617385·10^-5 eV/K )
Elementary charge e: 1.60217733·10^-19 C
Avogadro number N_A: 6.0221367·10²³ particles/mol
Speed of light c: 2.99792458·10⁸ m/s
Permeability of vacuum ₀: ₀ = 4 ·10^-7 T²·m³/J; 12.566370614·10^-7 T²·m³/J
Permittivity of vacuum ₀: ₀ = 1 / (₀ c²); 8.854187817·10^-12 C²/J·m
Fine structure constant: 1 / 137.0359895
Electron rest mass m_e: 9.1093897·10^-31 kg
Proton rest mass m_p: 1.6726231·10^-27 kg
Neutron rest mass m_n: 1.6749286·10^-27 kg
Bohr magneton _B: _B = e h / (4 m_e); 9.2740154·10^-24 J/T
Nuclear magneton _N: _N = e h / (4 m_p); 5.0507866·10^-27 J/T
Free electron g factor g_e: 2.002319304386
Free electron gyromagnetic ratio _e: _e = 2 g_e _B / h; 1.7608592·10¹¹ 1/s·T; _e / (2 ) = 28.024944 GHz/T
Electron magnetic moment _e: _e = -(1/2) g_e _B; -9.2847701·10^-24 J/T
Proton gyromagnetic ratio (H₂O) _p: 2.67515255·10⁸ 1/s·T; _p / (2 ) = 42.576375 MHz/T
Proton magnetic moment _p: 1.41060761·10^-26 J/T
Proton-electron ratios: m_p / m_e = 1836.152701; _e / _p = 658.2106881; _e / _p = 658.2275841 (protons in water)

Charge-to-mass ratio for the electron e / m_e: 1.75880·10¹¹ C/kg
Atomic mass unit amu: 1.66054·10^-27 kg
Bohr radius a₀: 5.29177·10^-11 m
Electron radius r_e: 2.81792·10^-15 m
Gas constant R: R = N_A k_B; 8.31451 m²·kg/s²·K·mol
Molar volume V_mol: 22.41383 m³/kmol
Faraday constant F: F = N_A e; 9.64846·10⁴ C/mol
Proton g factor (Landé factor) g_H: 5.585
Gravitational constant G: (6.673 +- 0.010)·10^-11 m³/kg·s² (CODATA); 6.67390·10^-11 m³/kg·s² +- 0.0014 % (Jens Gundlach, Univ. of Washington; from: Der Tagesspiegel 2000-05-08); (6.6873 +- 0.0094)·10^-11 m³/kg·s² (Schwarz et al., Science 282, 2230 (1998))
Acceleration due to gravity g: 9.80665 m/s²
Compton wavelength of the electron _c: _c = h / (m_e c); 2.42631·10^-12 m

FURTHER USEFUL CONSTANTS

Atomic energy unit Hartree: 1 Hartree = e² / (4 ₀ a₀); 1 Hartree = 2.625501·10⁶ J/mol (approx. 627.5 kcal/mol)

USEFUL CONVERSION FACTORS

NMR (Nuclear Magnetic Resonance)

Proton Larmor frequency: _p = _p / (2 ) B; _p = 42.5764 MHz/T (H₂O)

EPR (Electron Paramagnetic Resonance, ESR)

(Note that the use of brackets [ ] in the following expressions is not in accordance with standards which require the use of a slash, e.g. A/MHz.)

Electron Larmor frequency: _e = _e / (2 ) (g / g_e) B; _e [GHz] = 13.9962 g B [T]; g = 0.07144775 _e [GHz] / B [T]; g = 3.04199 _e [GHz] / _p [MHz]; B [T] = 0.0234872 _p [MHz]
Conversion of Units: 1 G = 0.1 mT; 1 T = 10 kG; 1 mT = 10 G; A [MHz] = 2.80249 (g / g_e) A [G]; A [MHz] = 28.0249 (g / g_e) A [mT]; A [MHz] = 13.9962 g A [mT]; A [MHz] = 2.99792·10⁴ A [cm^-1]; A [cm^-1] = 0.333564·10^-4 A [MHz]; A [cm^-1] = 4.66863·10^-4 g A [mT]

Source : http://www.chemie.fu-berlin.de/chemistry/general/constants_en.html

Mirrors and Lenses

2008-11-12T08:09:00.000-08:00

at my this simulation will a few/little explain about simulation that give elementary concept at lens deflextion, and mirror reflection

in my this simulation gives link its so that you can take it. but I give from link in origin

in this simulation also provided some concept question [of] and noteses important

you can select lens or mirror that will be seen its concept.

[screen/sail] capture like seen on the top

and for download

link this its

Refraction and total internal Reflection

2008-11-11T07:54:00.000-08:00

at my this simulation will a few/little explain about simulation that explain concept from deflextion

in my this simulation gives link its so that you can take it. but I give from link in origin

in this simulation also provided some concept question [of] and noteses important

you can arrange index each medium, and arrange its incidence angle.

[screen/sail] capture like seen on the top

and for download
link this

RELATIVITY & SPECIAL RELATIVITY

2008-11-10T09:43:00.000-08:00

RELATIVITY & SPECIAL RELATIVITY topics

In 1905, Einstein produced 3 seminal papers ( seminal means "pregnant with possibilities") one of which is his "Special Relativity".

Relative motion is the comparison of positions, velocities and accelerations as we move from one "frame of reference" to another.

Consider two cars approaching each other head on, one at 60 kmh^-1, the other at 40 kmh^-1. But these velocities are measured relative to the side of the road, the side of the road is the understood "frame of reference".

If we now move into one of the cars, say the 60 kmh^-1 one and make it our frame of reference; then the road goes backwards at 60 kmh^-1 and the other car comes towards us at 100 kmh^-1. We have changed our frame of reference, so the relative motions of ground and other cars to alter.

The relative motion statement in algebra of the above is

v_BrelA = v_BrelG - v_ArelG(1) ( "BrelA" means "B relative to A" , G is the ground )

To see this in 2D, go to the animation below of two ships, A and B passing relative to a lighthouse, the ground, G.

In two dimensions, we use vectors to add or subtract velocities to move from one frame to another. Our usual frame is considered "stationary". As a result, we see motion relative to that frame possibly moving "sideways" or "backwards". We see this when sitting in a car with other traffic around us. Our brain interprets the situation as normal - we don't think about it consciously.

The equation is assumed by us in everyday life and was by Newton. Indeed, the equation is a statement of "Newtonian relativity" (also called the Galilean velocity transformation).

Newtonian physics holds that the only things to stay the same when moving between inertial frames of reference are acceleration and Newton's Laws. ( Things that stay the same from frame to frame are called INVARIANTS. ) In Inertial Frames, no matter who you are or what you are doing, you will, when measuring a racing car's acceleration, get the same value as everyone else (providing you didn't stuff up the experiment)!

It is automatically assumed that momentum and energy laws are the same between inertial frames as they are apparently part of Newton's Laws. ( This turns out to be false.)

Non accelerating frames are "inertial" - Newton's Laws clearly apply, but accelerating frames are non-inertial and Newton's Laws seem to go wrong. When in a car is going around in a circle, it is accelerating, velocity is changing, a force SEEMS to push you outwards - the so called "centrifugal" force. The car is accelerating so non Newtonian Forces seem to exist. Step outside and look from the side of the road which isn't accelerating, and it is clear that an inward force exists on the car. Newton's Laws rule OK.

In Newtonian physics, all inertial frames - frames moving at a "constant velocity" - are the same and that an experiment carried out entirely in such a frame cannot tell the experimenter its absolute velocity, only its velocity compared to other inertial frames. Sit in a smoothly running carriage or plane and it seems motionless UNLESS you look outside. Is there a frame TRULY AT REST? Newtonian physics cannot find one. To Newtonian physicists, all inertial frames of reference are "relative".

THE TROUBLE WITH LIGHT

James Clerk Maxwell developed a set of equations in 1861 which actually PREDICTED the speed of light and modelled light in terms of electric and magnetic fields. This was a brilliant step forwards.

In the C19th, physicists only had mechanical models as their standard. They tended to think in terms of Newton's Laws, so to them, light should have a medium to flow through. As sound has air, light should have a "Luminiferous ether", and as air is to sound, the ether was to light. ( Note - "ethernet" is used today by computer nutters - how out of date is that! )

("Aethers" have a long history in physics - Descarte used vortices in an aether to describe - incorrectly- the planets' motions. The word means "an unearthly, delicate substance".)

Properties of the ether

1. It has to be inside any and every transparent material including glass and diamond - else why can light pass through?

2. It has to be very, very "stiff" - high rigidity - as otherwise waves will not move very fast in it!

3. It has to be so very, very thin that it causes no drag to motion, planetary or animal.

Good stuff isn't it.

The trouble is that it seems that this is an ABSOLUTE FRAME if we accept the equation (1) above. In otherwords, different people should see different speeds depending on what they are doing! The ether frame should be measurable by experiment by measuring the speed of light while travelling at different velocities.

So, physicists believed that electromagnetism had an Absolute Frame of Reference - the ether frame, but Newton's physics had NO absolute frame. A dilemma.

Attempts to detect the ether. Michelson.

Special Relativity