Coupled Resonant Pendulums - These pendulums trade swings back and forth

Coupled Resonant Pendulums

 

These pendulums trade swings back and forth.

Two pendulums suspended from a common support will swing back and forth in intriguing patterns if the support allows the motion of one pendulum to influence the motion of the other. 

 

• 2 plastic 35 mm film cans.
• Clay, coins, or washers for mass.
• 2 pieces of metal coat hanger wire, each about 8 inches (20 cm) long.
• A piece of string about 3 feet (90 cm) long.
• 2 ring stands or other vertical supports for the string.

(30 minutes or less)

Stretch and secure the string between two ring stands placed about 20 to 30 inches (50 to 75 cm) apart. In the center of each film can lid, punch a hole just large enough to insert one end of a coat hanger wire. Bend the end of the inserted wire so the lid won't slide off but so that you can still put the lid on the can. Bend the other end of the wire so it will hang freely from the string. The two hangers should be close to the same length. Add equal amounts of clay, coins, or washers to each can and attach the lids. Hang the pendulums so that they are about equally spaced from each other and from the ring stands.

(15 minutes or more)

Gently pull one pendulum back a short distance and let it go. As it swings back and forth, notice that the other pendulum also begins to move, picking up speed and amplitude with each swing. Notice that the pendulum you originally moved slows down with each swing and eventually stops, leaving the second pendulum briefly swinging by itself. But then the process begins to reverse, and soon the first pendulum is swinging again while the second one is stopped. The pendulums repeatedly transfer the motion back and forth between them this way as long as they continue to swing. Experiment with different wire lengths and with different string tensions to produce more strongly or weakly interdependent coupled pendulums.

Every pendulum has a natural or resonant frequency, which is the number of times the pendulum swings back and forth per second. The resonant frequency depends on the pendulum's length. Longer pendulums have lower frequencies.

Every time the first pendulum swings, it pulls on the connecting string and gives the second pendulum a small tug. Since the two pendulums have the same length, the pulls of the first pendulum on the second occur exactly at the natural frequency of the second pendulum, so the second pendulum begins to swing too. The second pendulum swings slightly out of phase with the first one. That is, when the first pendulum is at the height of its swing, the second pendulum is still somewhere in the middle of its swing. As soon as the second pendulum starts to swing, it starts pulling back on the first pendulum. These pulls are timed so that the first pendulum slows down. To picture this, it may help you to think of a playground swing. When you push on the swing at just the right moments, it goes higher and higher. When you push the swing at just the wrong moments, it slows down and stops.

The second pendulum pulls on the first pendulum at just the "wrong" moments. Eventually, the first pendulum is brought to rest; it has transferred all of its energy to the second pendulum. But now the original situation is exactly reversed, and the first pendulum is in a position to begin stealing energy back from the second. And so it goes, the energy repeatedly switching back and forth until friction and air resistance finally steal all of it away from both pendulums.

If the two pendulums are not the same length, then the tugs from the first pendulum's swings will not occur at the natural frequency of the second one. The two pendulums swing but with an uneven, jerky motion.

It is easy to predict how often the two swinging cans will trade energy. Count the total number of swings per minute when you start both pendulums together and they swing back and forth, side by side. Compare that to the number of swings per minute when you start them opposite one another - that is, with one pulled forward and one pulled backward an equal distance from the string, and then released at the same time. The difference between those two numbers exactly equals the number of times per minute that the pendulums pass the energy back and forth if you start just one pendulum while the other hangs at rest. Physicists call these two particular motions normal modes of the two pendulum system, and they call the difference between the frequencies of the normal modes a beat frequency.

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Corner Reflector - See yourself as others see you!

  

Corner Reflector

 

See yourself as others see you. 

 

Two hinged mirrors create a kaleidoscope that shows multiple images of an object. The number of images depends on the angle between the mirrors. When you set the hinged mirrors on top of a third mirror, you create a reflector that always sends light back in the direction from which it came. 

 

• Three 6 x 6 inch (15 x 15 cm) mirrors. Plastic mirrors are best, since there is less danger of breaking the mirror or cutting your fingers. Plastic mirrors are available at plastics supply stores and can easily be cut to any size. Glass mirror tiles are readily available but are not as safe.
• Duct tape.
• A piece of light cardboard (such as a manila file folder).
• Adult help.

(30 minutes or less)

If you start with one large piece of mirror, cut three 6 x 6 inch (15 x 15 cm) pieces from the plastic or glass. You can cut the plastic with a fine saw, such as a hacksaw, or score it with a utility knife and then snap it off. It is not hard to cut glass; get someone who knows how to do it to show you. (WARNING: For safety, after cutting a glass mirror, mount it on wood or cardboard and cover the edges with duct tape.)

Once you have the three mirrors you need, use the duct tape to tape two of the mirrors together along one edge. Put the tape on the back side of the mirror, making a hinge that opens and closes easily. Be sure the mirrors can move freely from 0 degrees to 180 degrees.

(15 minutes or more)

To make a kaleidoscope, set the hinged mirrors on the cardboard, and place an object such as a pencil or some coins between them. Open the mirrors to different angles. Notice that the smaller the angle, the greater the number of images you see. Remove the objects and see what happens when you draw different designs in the space between the two mirrors.

Close your right eye and look at a single mirror straight on. Notice that the left eye of the image is closed. Now close your right eye and look at two mirrors that form a 90-degree angle. Notice that the right eye of this image is closed.

Now make a corner reflector by opening the two taped mirrors to 90 degrees and resting them on the third mirror, so that the three mirrors form a half cube (see diagram). Close one eye and stare right at the corner where the three mirrors join. Move your head and notice that the pupil of your open eye always falls right at the corner. Open both eyes and look at the corner. One eye may appear to be closer to the corner than the other. This is your dominant eye.

When you put an object between the two hinged mirrors, light from the object bounces back and forth between the mirrors before it reaches your eyes. An image is formed each time the light bounces off a mirror. The number of images that you see in the mirrors depends on the angle that the mirrors form. As you make the angle between the mirrors smaller, the light bounces back and forth more times, and you see more images.

The illustration below shows how an image is formed in the corner of two mirrors at 90 degrees. Light rays bounce off each mirror at the same angle that they hit the mirror: Physicists say that the angle of reflection is equal to the angle of incidence. Mirrors at other angles behave similarly, but the ray diagrams may get more complex.

The inside corner of a corner reflector (where the three mirrors meet) sends light back parallel to its original path. If you pointed a thin beam of laser light right near the corner, the beam would bounce from mirror to mirror and then exit parallel to the entering beam. Light from the center of your eye bounces straight back to the center of your eye, so the image of your eye seems to be centered in the corner made by the mirrors.

In a corner reflector, multiple reflections reverse the image and invert it.

Corner reflectors are used to make safety reflectors for cars, bicycles, and signs. Corner reflectors have also been used to bounce laser beams back to the earth from the surface of the moon.

Throw a tennis ball into the corner of a room. It should return to you after bouncing off the three surfaces.

Tape five square mirrors together with the mirrored surfaces facing inward to form a box. Place a sixth mirror, turned at a 45-degree angle, over the open side so you can look into the box and also let some light in. This combines the Look into Infinity Snack with this Corner Reflector Snack. Try other configurations of mirrors in three dimensions and see what you can discover.

To do a quantitative experiment, mark the following angles on a piece of cardboard: 180 degrees, 90 degrees, 60 degrees, 45 degrees, 36 degrees, 30 degrees, and 20 degrees. These angles are chosen so that when they are divided into 360 degrees they produce an even integer. Mount the hinged mirrors at each of these angles and place an object betvveen them. Count the number of images you see. You should be able to verify the following rule: 360 divided by the angle between the mirrors gives the number of images, plus one. At 60 degrees, for example, 360/60 = 6, so you should see five images of the object.

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Cool Hot Rod - Objects change size when heated or cooled

Cool Hot Rod

 

Objects change size when heated or cooled 

 

Changes in temperature cause objects to expand or contract. You may have noticed the effects of this kind of change around your house. If you run cool water on a hot glass, it may break, as some parts of the glass contract more rapidly than others. You can loosen a tight jar lid by running hot water over it, causing the metal lid to expand more than the glass. The expansion and contraction of materials when they are heated or cooled is commonly used to make thermometers and thermostats. This Snack allows you to directly observe the expansion and contraction of a metal tube. 

 

• 3 to 6 feet (90 to 180 cm) of straight l/4 inch (6 mm) copper tubing.
• A small funnel.
• 1 foot (30 cm) of plastic tubing. Choose a size that will fit snugly over the end of the copper tubing and over the end of the funnel.
• A ring stand.
• A "C" clamp.
• A bucket.
• 2 small blocks of smooth wood.
• A large needle.
• A toothpick.
• Hot water.
• Cold water.
• Adult help.

(one hour or less)

Insert one end of the copper tubing into the plastic tubing. Slip the plastic tubing over the end of the funnel. Make sure that the tubes all fit snugly together.

Place the ring stand near the edge of a table. Position the ring so that it supports the funnel a few inches above the tabletop. Position the copper tubing so that the end farthest from the funnel sticks out beyond the edge of the table by a few inches. Place a small block under the copper tubing at the end near the funnel. Clamp the tubing and block to the table so that they can't move.

Place the second block under the other end of the tubing. Put the needle between the copper tubing and the block, positioned perpendicular to the tubing. Make sure that the eye of the needle extends past the block. Stick the toothpick through the eye of the needle. As the tubing expands and contracts, the needle will rotate, rolled along by the movement of the tubing. The toothpick will shift from an upright position to a slanted position as the needle rotates, making the rotation more evident.

Put the bucket under the end of the copper tubing that sticks out beyond the edge of the table. The bucket will catch the water that you will pour through the copper tubing. 
 

(15 minutes or more)

Pour hot water into the funnel to heat the tubing. For best results, heat the water to near boiling. When you do this, remember to keep your hands away from the copper tubing: It will become very hot!

When you pour the hot water into the funnel, notice the direction in which the needle rotates. Immediately pour cold water through the funnel, and watch the needle again. Notice the direction in which it rotates. 
 

The copper tubing, like everything else in the world, is made of atoms that are constantly vibrating. The higher the temperature, the faster the atoms vibrate. When you pour hot water into the tubing, heat flows from the water to the copper, giving energy to the copper atoms, which vibrate faster. This increase in vibration causes the atoms to collide with each other more often and more violently, so the space between the atoms increases. As a result, the whole tube gets longer and thicker. The needle turns as the tube expands.

When you pour cold water into the tube, the copper atoms give up some of their heat energy to the water, vibrate less violently, and move closer together. The tube shrinks and the needle turns in the opposite direction as the tube contracts.

The copper tube expands by 1.7 x 10^-5 of its length for every 1.8 degrees Fahrenheit (1 degree Celsius) of temperature increase. So a copper tube that is 3.3 feet (1 meter) long will expand by 5.6 x 10^-3 feet (1.7 x 10^-3 m) over a 180°F (100°C) temperature change, lengthening by almost 0.06 inch (1.7 mm). As the copper tube expands, it will make the needle roll over this 0.06 inch (1.7 mm) distance. When an average-sized needle rolls 0.06 inch (1.7 mm), it makes more than two complete revolutions. The toothpick in the eye of the needle dramatically amplifies the motion of the expanding or contracting rod.

If the needle slips instead of rotating, try placing a microscope slide between the wood and the needle. You can also increase the friction by wrapping a rubber band around the wood and the tubing to hold them together more tightly.

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Convection Current - Make your own heat waves in an aquarium

   

Convection Current

 

Make your own heat waves in an aquarium.

This demonstration gives you a simple and visually appealing way to show convection currents in water. Warmer water rising through cooler water creates turbulence effects that bend light, allowing you to project swirling shadows onto a screen. 

 

• One 6- or 12-volt lantern battery or a suitable low-voltage battery eliminator or power supply.
• A pencil lead.
• A clear plastic or glass container with rectangular flat sides. (A small aquarium works fine.)
• A light source. (Slide projectors, filmstrip projectors, or flashlights work well. A point-source flashlight, such as a MiniMaglite™ flashlight with the reflector removed, will produce sharp images.)
• Food coloring (in a small dropper bottle or an eyedropper).
• A projection screen or white posterboard.
• 2 electrical lead wires with alligator clips at both ends (available at Radio Shack).
• Tap water.
• Optional: Switch or dimmer switch (both available at hardware stores) or any sort of rheostat or variable resistor.
• Adult help.

(15 minutes or less)

Use one clip lead to attach the positive terminal of the battery to one end of the pencil lead, and the second clip lead to attach the negative terminal to the other end of the pencil lead. If you like, you may connect a simple switch, or a dimmer switch, in series. The switch makes using the device more convenient; the dimmer switch lets you vary the amount of current going through the carbon rod.

Fill the container with water and place the wires and pencil lead in it so that the pencil lead is positioned horizontally. Connect the two wires to the terminals of the battery, and allow the heating to start. Shine the projector through the liquid, projecting the light onto the screen or white posterboard.
 

(15 minutes or more)

Observe the convection currents. If you have a dimmer switch, vary the current and observe the effects of the various settings. If you are using a rheostat or variable resistor, you may have to try several settings to find which one works best. You can also vary the orientation of the pencil lead to see if this has any significant effect on the convection pattern. Add a few drops of food coloring and observe the effects.

Like air, water expands as it gets warmer and so becomes less dense. Since the water warmed by the current flowing through the carbon rod is less dense than the surrounding colder water, the warm water rises through the colder water to the surface, causing the food coloring to move along with it.

Since the cold and warm water have different densities, they have different indices of refraction. Light bends (refracts) as it passes from warmer to colder or colder to warmer. When light is bent onto an area of the screen, that area becomes brighter. When light is bent away from an area of the screen, that area becomes darker. The positions of warm and cold water are constantly changing, so the images projected on the screen shimmer and flow like heat waves in air.

A simpler method of doing this Snack, which allows people to perform it themselves, is simply to place a candle on a table and project this image onto a screen with a flashlight. The point source of a MiniMaglite™ projects clear images of convection when used on a small-scale desktop experiment. Changing the distance from a point light source to the candle will change the magnification of the image of the convection currents projected on the wall.

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Condiment Diver - The world's simplest Cartesian diver

 

Condiment Diver

 

The world's simplest Cartesian diver

Squeezing a plastic bottle filled with water and a condiment packet makes the packet sink. Letting go of the bottle makes the packet rise. 

 

• Squeeze condiment packet (soy sauce, ketchup, etc.)
• Clear plastic bottle with tight-fitting lid
• A glass or cup of water

First, you have to figure out if your condiment packet is a good Cartesian diver candidate. Fill a glass with water and drop in your packet. The best packets are ones that just barely float. 

After you have found the proper packet, fill an empty, clear plastic bottle to the top with water. Shove your unopened condiment packet into the bottle. Replace the cap... and you're done! Squeeze the bottle to make the diver sink, and release to make it rise. Amazing!

Many sauces are denser than water, but it is the air bubble at the top of the sauce that determines whether the packet will sink or swim. Squeezing the bottle causes the bubble to shrink. This smaller bubble is less buoyant and the packet sinks.

By Eric Muller
Orginally published in The Physics Teacher, May 1996

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