Section 4.1 Bicycles

Description: A plastic track is first bent to form a valley and a marble rolls into the bottom of the valley. The track is then bent to form a mountain and a marble is carefully balanced on top of the peak but rolls off it at the slightest disturbance. Finally, the track is made flat and level, and the marble is remains wherever it's left along this track.

Purpose: To show the different types of equilibriums.

Procedure: Use the blocks to bend the plastic track into a valley and put the marble at the bottom of the valley. Point out that the marble is in equilibrium because it's not accelerating and thus must be experiencing zero net force. Then disturb the marble and show that it always rolls back to the bottom of the valley—it's in a stable equilibrium.

Now rebend the track so that it forms a mountain rather than a valley. Carefully balance the marble on the top of the mountain. It will again be in equilibrium, but now this equilibrium is unstable. Show that the slightest disturbance of the marble will start it rolling down the mountain.

Finally, place the track flat and level on the table. The marble will now remain in equilibrium no matter where you put it along the track—it's in a neutral equilibrium.

Explanation: A stable equilibrium is one to which the displaced object will return when released—the displaced object experiences a restoring force that pushes it back toward the equilibrium position. An unstable equilibrium is one to which the disturbed object won't return when released—the displaced object experiences a force that pushes it away from the equilibrium position. An object that is displaced from a neutral equilibrium experiences no force either toward or away from the original equilibrium position.

Description: A tricycle is stable against tipping while stationary.

Purpose: To illustrate the static stability.

Procedure: Sit on the tricycle (assuming that you can fit and that your pride will allow you to do so). Show that as long as your center of gravity remains over the base of support (within the triangle bounded by the three wheels), the bicycle is in a stable equilibrium—after small tips, it will return to its original situation. Point out that this recipe for static stability (center of gravity above base of support) is a special case of the general physics rule for a stable equilibrium: that any disturbance from a stable equilibrium raises the total potential energy. That's why the equilibrium is stable: there are restoring influences that return you to the equilibrium.

Explanation: Tipping the stationary tricycle raises its center of gravity, so it experiences a restoring force when tipped and naturally returns to its upright orientation. An object accelerates in whatever fashion lowers their total potential energy as quickly as possible, so the tipped tricycle returns to its upright equilibrium.

Description: When you sit in the middle of a chair, it remains upright, even as you jiggle about. But if you lean far away from the center of the chair and its base of support, you and the chair will tip over.

Purpose: To show that an object will tip over when its center of gravity is no longer over its base of support.

Procedure: Sit in the chair and roughly locate the center of gravity of you and the chair. Show that tipping the chair slightly won't cause it to tip over because the overall center of gravity will rise and you and the chair will naturally accelerate back toward your original situation—you and the chair are in a stable equilibrium. Now begin to lean far out to one side until the overall center of gravity is no longer above the base of support (the area bordered by the chair's legs). The chair will tip over.

Explanation: When your center of gravity is above the edge of the base of support, you're in an unstable equilibrium -- a maximum of total potential energy -- and you'll accelerate away from that unstable equilibrium whenever disturbed from it. As long as the overall center of gravity is above the chair's base of support, tipping the chair raises the center of gravity. But once you begin leaning and the overall center of gravity is no longer above the chair's base of support, tipping the chair lowers the center of gravity. It begins to tip farther and farther so as to lower its gravitational potential energy.

Description: An object's center of gravity is slowly shifted until it's no longer above the object's base of support. The object then tips over.

Purpose: To show that when an object's center of gravity can descend as it tips, that object will tip over.

Procedure: Start with the demonstrator's center of gravity located above the demonstrator's base of support. Show that no matter which direction you tip the demonstrator, its center of gravity rises. Point out that because there is no direction in which the demonstrator can tip and lower its gravitational potential energy, the demonstrator will not tip. It's in a stable equilibrium—it naturally returns to its original position after begin tipped slightly.

Now shift the demonstrator's center of gravity gradually until it's no longer above the demonstrator's base of support. It will tip over. Point out the once the demonstrator's center of gravity is outside its base of support, there is a direction in which the demonstrator can tip and lower its center of gravity. Since an object accelerates in the direction that lowers its potential energy as quickly as possible, the demonstrator accelerates in the direction that lowers its center of gravity, and its gravitational potential energy, as quickly as possible.

Explanation: While gravity really acts throughout the demonstrator, it effectively acts at the demonstrator's center of gravity (which coincides with the demonstrator's center of mass). If that center of gravity rises, then the demonstrator's gravitational potential energy rises and if that center of gravity falls, then the demonstrator's gravitational potential energy falls. Since objects accelerate in the direction that reduces their potential energy as quickly as possible, the demonstrator will only tip over if doing so will reduce its gravitational potential energy. Thus it will only tip over if doing so will immediately lower its center of gravity. For geometrical reasons, that lowering will occur only when the center of gravity is not above the demonstrator's base of support.

Description: A bicycle is unstable against tipping while stationary

Purpose: To show the lack of static stability in a two-wheeled vehicle.

Procedure: Sit on a bicycle and show how difficult it is to keep it upright while it's stationary. Unless you are extremely talented, you will be unable to keep it upright for more than a second or two. The bicycle is statically unstable.

Explanation: When you and the bicycle are stationary, your overall center of gravity descends whenever the bicycle tips to one side. As a result, the upright bicycle is in an unstable equilibrium and tips over at the slightest perturbation.

Description: A tricycle tips over during a sharp turn at high speed.

Purpose: To illustrate that statically stable objects can have dynamic instabilities.

Procedure: Ride the tricycle rapidly across the floor and make a sudden sharp turn. The tricycle will tip over (be prepared to stop yourself from falling so that you don't get injured). Note that while the tricycle is statically stable, it's dynamically unstable.

Explanation: When the tricycle is moving and you execute a sudden sharp turn, the friction between the ground and the wheel not only causes the tricycle to turn in the direction you want, it also exerts a torque on the tricycle about its center of mass. If this torque is large enough, the tricycle's static stability isn't enough and the tricycle begins to rotate. The wheels rotate in the direction of the turn and your head rotates in the opposite direction. You and the tricycle tip over. Viewed in another way, you travel in the straightline path that inertia dictates while the tricycle accelerates out from under you.

Description: A bicycle can avoid tipping over even during a sharp turn at high speed.

Purpose: To illustrate that statically unstable objects can have dynamic stabilities.

Procedure: Ride the bicycle forward across the floor and show how easy it is to keep it upright while it's moving forward. Make a turn (if you have room) and show that by leaning into the turn, you keep it from tipping over. This holds true no matter how fast you are going or how sharp the turn, as long as the wheels don't skid across the ground. Note that while the bicycle is statically unstable, it's dynamically stable.

Explanation: When you and the bicycle are moving, at least two factors contribute to the bicycle's dynamic stability (the next demonstrations). Moreover, during a turn, you lean into that turn and the overall force exerted on the wheel by the ground—a combination of an upward support force and a horizontal frictional force—points directly at the overall center of mass. As a result, you and the bicycle don't begin to rotate and the bicycle remains stable throughout the turn. Viewed in another way, you are balancing two separate torques: the torque that ground friction exerts on the wheel (the one that flipped the tricycle) with the torque that the ground support exerts on the wheel (the one that tips over a stationary bicycle).

Description: You balance a broom on your hand by always putting your hand under its center of gravity.

Purpose: To illustrate that statically unstable objects can be balanced by always returning them to their unstable equilibriums.

Procedure: Stand the broom up in your hand and then keep it balanced by moving your hand under its center of gravity each time it tips. Note that you must keep your eyes on the broom's center of gravity because that's what you're putting your hand under. If you try to gaze at your hand or at the tip of the broom, you won't be able to balance the broom.

Explanation: When the broom is upright, it's in an unstable equilibrium. If you don't act to help it stay that way, it will fall over. But by always returning it to the unstable equilibrium whenever it tips, you can keep it upright indefinitely.

Description: You straddle the upright bicycle and then tip it to its left. The front wheel spontaneously turns toward the left so that the bicycle is poised to drive under your center of gravity if you were moving forward.

Purpose: To show one of the origins of the bicycle's enormous dynamic stability.

Procedure: Straddle the upright bicycle and make sure that its front wheel points straight ahead. Point out that if you were to sit on the seat that it would be in an unstable equilibrium—your overall center of gravity would be directly above the line between the two wheels; a minimal base of support. Now align the 2-meter stick with the stem of the front wheel and show that the axis of rotation about which the front fork turns intersects the ground in front of the contact point between the front wheel and the ground. Note that this arrangement causes the front wheel to steer toward the left as the bicycle tips toward the left. Now tip the bicycle to its left and show that the front wheel steers toward the left. Point out that this steering effect causes the bicycle to drive itself under your center of mass and thus returns the bicycle to its unstable static equilibrium. As long as the bicycle is moving forward, the unstable equilibrium is actually a stable equilibrium; a dynamically stable equilibrium.

Explanation: If dynamical processes act to return an object to its unstable static equilibrium, then that equilibrium is dynamically stable. In this case, the automatic turning of the front wheel returns the moving bicycle to its upright position—an unstable static equilibrium.

Description: A gyroscope mounted in gimbals acts like the front wheel of a bicycle, turning in the direction the wheel would when the bicycle is rolling forward. When a weight is attached to the left side of the gyroscope's axle—simulating a lean to the left—the gyroscope/wheel begins to turn to the left. This gyroscopic precession would cause the bicycle to drive under the rider's center of gravity and would return the bicycle to its upright, unstable static equilibrium.

Purpose: To show how an unstable static equilibrium can be made stable by dynamic processes and to demonstrate gyroscopic precision.

Procedure: Spin the gyroscope in such as way that it resembles the front wheel of a bicycle moving forward. You might prop a bicycle next to it so that everyone can see that the gyroscope wheel is turning in the direction that the front wheel of the bicycle would turn. Now hang the weight from the left side of the axle supporting the gyroscope wheel. The gyroscope wheel will turn toward the left—it will precess. This direction of turning would cause a bicycle moving forward to drive itself under the rider's center of gravity and return the bicycle to its upright unstable static equilibrium orientation. Thus the front wheel's gyroscopic character causes the bicycle to be dynamically stable.

Explanation: When the gyroscope wheel is spinning like the front wheel of a bicycle (its top surface is turning forward and away from you when you are standing as though you were riding the bicycle), the wheel's axis of rotation points horizontally to your left (according to the right hand rule). When you add the weight to the left side of the axle, it exerts a torque on the gyroscope that points horizontally rearward (toward the rear wheel of the hypothetical bicycle). The resulting angular acceleration slowly shifts the gyroscope's axis of rotation from leftward to rearward—i.e. the wheel turns toward the left.

Description: You turn the pedals of an inverted bicycle by hand. Changing the gears allows you to vary the number of turns the rear wheel makes for each complete turn of the pedals.

Purpose: To show that gears, belts, and chain drives allow you to vary the mechanical advantage between two rotating systems.

Procedure: Flip the bicycle upside down and prop it on its handlebars. Turn the pedals by hand and show that the number of turns that the rear wheel makes for each turn of the pedals depends upon the choice of gears. With the chain going around the largest crank sprocket and around the smallest freewheel (rear wheel) sprocket, one turn of the pedals causes the rear wheel to turn several times. With the chain going around the smallest crank sprocket and around the largest freewheel sprocket, one turn of the pedals causes the rear wheel to turn only about one time. Discuss how, in the first case, your pedaling exerts relatively little torque on the rear wheel—which is why it's difficult to climb a hill in this gear; the frictional torque on the rear wheel stops it from turning. Also discuss how in the second case, your pedaling exerts a relatively large torque on the rear wheel—which is why its easy to climb a hill in this gear; the frictional torque on the rear wheel is easily overcome by pedaling. Of course, you don't get something for nothing: in the hill climbing gear, you must turn the pedals many times to make any reasonable progress up the hill. You're doing the work needed to lift yourself up the hill by exerting modest forces on the pedals but making them move long distances (many turns) in the directions of those forces.

Explanation: The gear system on the bicycle provides mechanical advantage. When you are on level pavement, you need relatively little torque to turn the rear wheel. By using a large crank sprocket and a small freewheel sprocket, you can make the rear wheel turn as rapidly as possible. Your pedaling exerts relatively little torque on the rear wheel but that rear wheel turns many times for each turn of the pedals. When you are climbing a hill, you need a relatively large torque to turn the rear wheel—friction with the ground is acting to slow the wheel's rotation. By using a small crank sprocket and a large freewheel sprocket, you can exert a large torque on the rear wheel although it will turn relatively slowly. You must turn the pedals many times to climb the hill a reasonable distance.

Description: Lights show that a wheel's rim moves at different speeds from the rest of the bicycle.

Purpose: To show why it's important to keep the mass of the tires low.

Procedure: Attach one flashlight to the center of the disk and the other flashlight to the disk's rim. Turn on the flashlights and dim the room lights. Roll the wheel across a long table or the floor and observe the lights. The light at the center of the wheel will move steadily forward, but the light at the wheel's rim will execute cycloidal motion. It will briefly stop moving altogether when it's at the bottom of the wheel and it will move forward at twice the speed of the wheel's center when it's on the top of the wheel. While the rim light's average speed is still equal to that of the center light, the rim light's average kinetic energy is twice that of the center light.

Explanation: Kinetic energy depends on the square of speed. Although the rim light's average speed is equal to that of the center light, the rim light spends half its time moving at more than this average speed and its kinetic energy during that time is quite large. When the rim light is at the top of the wheel and moving forward at twice the speed of the center light, its kinetic energy is four times as large as that of the center light. Overall, the rim light's average kinetic energy is twice as large as that of the center light. Because you must provide the work that gives this increased energy to the tire of your bicycle, you want as little mass in that tire as possible.

Description: A student sits motionless on a cart with a heavy medicine ball in his lap. When he throws the ball in one direction, the car begins to roll in the opposite direction.

Purpose: To show that the act of pushing an object away causes him to accelerate in the opposite direction and to show that, while the momentum of an isolated system can't change, that momentum can be redistributed among the pieces of the system.

Procedure: Have the student sit motionless on the cart with the heavy ball in his lap. Point out that his present momentum, including the cart and ball, is exactly zero. Discuss what will happen when he throws the ball in one direction—how he will have to push the ball away from him, how it will accelerate in the direction of his push, and how it will end up with momentum in that direction. Note also that it will push back on him, that he will accelerate in the direction of its push on him, and that he will end up with momentum in the direction of its push. Point out that the total momentum of the student, the cart, and ball will still be zero, but that it will now be rearranged. At this point, have the student throw the ball as hard as he can in one direction and he will begin rolling in the opposite direction (make sure that he throws the ball opposite a direction the cart can roll). Point out that the ball didn't have to hit anything for him to begin moving. The very action of pushing it away was all that was needed—it pushed on him as he pushed on it.

Explanation: The student rolls in the direction opposite to the ball's motion because it pushes on him as he pushes on it. The ball didn't have to hit anything for him to accelerate in the opposite direction.

Description: A balloon attached to a cart on an air track accelerates in one direction as the air it contains accelerates in the opposite direction.

Purpose: To show that the act of pushing gas in one direction causes the device pushing that gas to accelerate in the opposite direction.

Procedure: Attach the balloon to the nozzle and then tape the nozzle to the top of the air track cart. Inflate the balloon, pinch it closed, and put the cart on the operating air track. Without pushing the cart, release the cart and balloon. As the balloon deflates, the cart will accelerate in the direction opposite the air stream.

Explanation: The balloon squeezes the air out of the nozzle and pushes that air in one direction. The air pushes back on the balloon and this reaction force causes the cart to accelerate in the other direction.

Description: You (or a student) sit on a cart with a modified carbon dioxide fire extinguisher attached to it. When you squeeze the release lever, a jet of gas emerges in one direction and you rocket across the room in the other direction.

Purpose: To show that pushing a stream of stored gas in one direction produces a force of equal magnitude in the opposite direction.

Procedure: Unscrew the conical diffuser from the carbon dioxide fire extinguish and expose the outlet holes that are connected to the main valve. The gas flowing out of the main valve hits the end of this outlet structure and then turns to flow in all directions through a set of six outlet holes. Carefully! cut off the very end of this outlet structure—just the last fraction of a centimeter—so that the gas no longer turns to flow out of the holes. Without the surface at the end of this structure to deflect the gas flow sideways, the jet of gas leaving the main valve will travel in a straight path at enormous speed. Under no circumstances should you ever cut into the fire extinguisher anywhere but after the main valve! Safety first!

If you have a pipe bolted to the cart, insert the modified fire extinguisher into it and wedge the fire extinguisher as necessary so that it will remain in the pipe when gas is flowing out of it. Then sit on the cart. If you don't have a pipe to hold the fire extinguisher, sit on the cart and hold the fire extinguisher tightly in your lap. It will push rather hard when gas is flowing out of it, so you should be prepared to hold on to it and to stop releasing gas if you find the reaction force uncomfortable.

Once you are seated on the cart and the fire extinguisher is pointed along a direction in which the cart can roll, squeeze the release handle and allow the gas to flow. When you're sure that you can handle the reaction forces, squeeze the handle completely so that the flow of gas is vigorous. The gas will stream out in a roaring torrent of white "smoke" and you and the cart will accelerate in the opposite direction. Be prepared to stop releasing gas and to stop your motion before you hit anything. You may want to have someone to "catch" you before you crash, just in case. A full fire extinguisher will last about 6 to 10 seconds at full thrust. I can usually get 2 trips across the front of the lecture hall before running completely out of gas.

Explanation: As the carbon dioxide in the fire extinguisher turns the corner in the main valve and accelerates out of the modified opening, the valve structure pushes on it and it pushes back. This reaction force causes the fire extinguisher, the cart, and you to accelerate in the direction opposite the gas flow.

Description: A toy water rocket is partly filled with water and attached to the launcher. After pumping air into the rocket, the rocket is released. It flies into the air as it ejects a stream of water in the opposite direction.

Purpose: To show that ejecting a stream of water from the exhaust nozzle of a rocket can cause that rocket to accelerate in the opposite direction.

Procedure: Partially fill the water rocket with water, according to the instructions. Attached the rocket to the launcher and pump air into the region of the rocket above the water. Describe what will happen when you release the rocket—the compressed air will push the pressurized water downward through the rocket nozzle. The water's pressure potential energy will become kinetic energy as it flows through the narrow nozzle and the water will leave the rocket at atmospheric pressure but with a large downward velocity. The rocket will have pushed the water downward to give it this velocity and the water will have pushed back, lifting the rocket into the air. Note that ejecting water from the rocket is more effective than ejecting air because the water has more mass and is harder to accelerate. The rocket pushes harder on the water than it would on air and the water pushes back harder on the rocket. You can make a similar analysis in terms of momentum—the water carries away more momentum because of its greater mass.

Explanation: As the rocket pushes the water downward and that water accelerates downward, the water pushes the rocket upward and the rocket accelerates upward. While gravity introduces an additional force which causes the entire system's center of mass to fall, the rocket's upward acceleration is so great that it rises into the air.

Description: An aluminum foil-wrapped match, tipped almost upright against a bent paper clip, is heated with another match until it ignites. The jet of gas flowing out from under the aluminum foil and heading down the match stick pushes on the match and sends it flying into the air.

Purpose: To show that when a fuel burns to form a high pressure gas and this gas flows in one direction, the gas's container accelerates in the opposite direction.

Procedure: Cut a small piece of aluminum foil and wrap two layers of it around the head of a match. The aluminum foil should extend about 5 mm beyond each end of the match head. Fold the free end of the aluminum foil to seal it—the only direction in which gas should be able to flow is down the stick. Now bend the paper clip so that it forms a prop for the match. The match should be about 20° from vertical. Now light the other match and carefully bake the aluminum wrapper with the flame. When the wrapped match head ignites, the torrent of gas flowing down the stick should push the match into the air. Experiment with different types of matches or wrapping techniques to obtain the best results.

Explanation: When the wrapped match head burns, it creates a large volume of hot gas that flows down the stick. The aluminum foil pushes the gas down the stick and the gas pushes back, propelling the match into the air.

Description: A plastic soda bottle with a drinking straw taped to its side rides an a long, taut string. After you spray a little hairspray into a hole drilled in the bottle's cap, you light it and the bottle propels itself along the string at a great speed.

Purpose: To show that combustion-heated gases can propel a rocket.

Procedure: Tape the plastic straw to the side of the bottle securely. Drill a 1/4" hole in the plastic cap of the bottle. Dry the insides of the bottle thoroughly and screw the cap on securely. Thread the string through the straw and tie the ends of the string to stable objects so that the string is nice and taut, and the string's destination end should be higher than the launch end. The bottle should start at the lower end of the string, with its sealed end pointing toward the destination end. Put on your safety glasses. To launch, spray the hairspray into the bottle through the hole in its cap. About 1 second of spray should be enough, although you'll have to experiment on your own. Light the match or spark lighter and carefully hold it against the hole in the bottle cap. Stand to the side of the bottle as you do this, not behind it -- the bottle cap can occasionally blow off and could hit people standing behind the bottle! If the mixture of air and hairspray inside the bottle is right, it will ignite and spray hot gas out of the hole in the bottle cap. The bottle will zoom along the string to its far end and then slide slowly back to its starting point. To reuse the rocket, assuming that the bottle didn't melt too much, you have to blow the burned gases and moisture out of the bottle, using compressed air or something equivalent. Otherwise, you just have to wait for fresh air to diffuse into the bottle. 10 seconds of compressed air usually does the trick nicely, allowing several launches before the bottle is too deformed for further use.

Explanation: When the hairspray and air burn, they create hot, high-pressure gases inside the bottle. These gases spray out the hole at very high speeds and carry backward momentum away from the bottle. The bottle thereby obtains forward momentum and travels up the string.

Description: A set of metal vanes, resembling a whirlybird water sprinkler, is attached to a static electric generator. As the voltage of the static generator builds, the metal vanes begin to spin.

Purpose: To show that pushing ions in one direction causes them to push back, propelling the source of those ions in the direction opposite the ion's velocity.

Procedure: Set the vanes on the metal pin bearing so that they can turn freely. Use the wire to connect the metal pin to the static generator. Now turn on the static generator so that charge begins to flow onto the vanes. When the charge becomes large enough, the vanes will begin to turn in the direction opposite the direction in which the sharpened points are directed.

Explanation: As charge accumulates on the vanes, it begins to leave the sharpened points and flow onto passing air molecules. These molecules become charged and are repelled by the remaining charges on the vanes. The ionized air molecules accelerate away from the points and the points accelerate away from the ionized air molecules. The vanes turn as the result of torques from these repulsions.

Description: You inflate a balloon and release it. While it experiences substantial thrust from the gas it's ejecting, it travels in random directions because it's aerodynamically unstable.

Purpose: To show the importance of aerodynamic stability for a rocket.

Procedure: Inflate the balloon and release it. It will fly around randomly until it runs out of air.

Explanation: The balloon's center of aerodynamic pressure is located near its front while the forward thrust of its gaseous exhaust is exerted near its rear. The front of the balloon tends to be slowed by air drag and the back of the balloon tends to be sped up by the thrust. As a result, the balloon tends to turn around in flight, over and over, and wanders almost randomly about the room until its air is used up.

Follow-up: Repeat the stability demonstrations from Section 4.3: the badminton birdie in flight and the arrow in flight. Early rockets used the same techniques to ensure that they flew nose first, tail last.

Description: You balance an arrow without feathers on a string and swing it around your head. At best, it is indifferent to which part is pointing forward. It may even tumble.

Purpose: To show that an featherless arrow is aerodynamically unstable.

Procedure: Tie the string around the arrow at its center of mass. The arrow hanging from the string should balance. Now swing the arrow around in a horizontal circle above your head. It will travel with random orientation and may even tumble.

Explanation: With no aerodynamic difference between front and back, the arrow has no torques to orient it in flight. It flies with whatever part forward it likes.

Description: You balance an arrow with feathers on a string and swing it around your head. It quickly orients itself tip forward and flies stably around your head.

Purpose: To show that an feathered arrow is aerodynamically stable.

Procedure: Tie the string around the arrow at its center of mass. The arrow hanging from the string should balance. Now swing the arrow around in a horizontal circle above your head. It will travel with point forward and feathers rearward.

Explanation: The feathers shift the center of aerodynamic pressure (the location of the air's overall force on the arrow) toward its feathers. The arrow can then experience a torque about its center of mass, one that restores the arrow to its desired point-first, feathers-last orientation.