Back to the playground.
A group of children are playing on a merry-go-round.
1. Compare the torque required to accelerate the merry-go-round to the same final angular velocity for the following two cases: (a) when the children are all at the outside edge of the merry-go-round and (b) when the children are all near the center of the merry go round.
Answer: The torque required when the kids are on the edge is larger than the torque required when they are near the center.
Why: The moment of inertia is larger when the kids are at the edge than when they are near the center. Angular acceleration is equal to the ratio of the torque to the moment of inertia. To produce the same angular acceleration less torque is required for the smaller moment of inertia.
2. The merry-go-round is spinning freely and all of the children are near its outside edge. Suddenly they all lean in toward the center of the merry-go-round. How does its angular velocity change? Why?
Answer/Why: The angular velocity increases. Since angular momentum must be conserved and is proportional to the product of moment of inertia and angular velocity, if the moment of inertia decreases the angular velocity must increase.
3. The merry-go-round is again spinning freely and all of the children are near its outside edge. Suddenly one of them lets go (without pushing off of the merry-go-round). She will of course accelerate toward the ground due to gravity, but what will the horizontal component of her velocity be (descriptive answer please)?
Answer: Her horizontal velocity will be exactly the same as just before she let go and will remain constant. It will have a nonzero amount (magnitude) and be in the direction perpendicular to the line between the axis of rotation of the merry-go-round and where she let go.
Why: The merry-go-round was providing a horizontal force pointing toward its pivot, which caused her to move in a circle. After letting go of the merry-go-round she no longer experiences this force and so the horizontal component of her velocity will remain constant.
4. Compare her kinetic energy immediately before and after letting go. If she lands on wet grass (no friction) what will her kinetic energy be after landing?
Answer: Her kinetic energy will be the same immediately before and after letting go. After landing on the grass her kinetic energy will again be the same as immediately before letting go.
Why: If she had pushed off of the merry-go-round it would have done work on her and her kinetic energy would have changed. Since she did not push off her velocity will be the same immediately before and after letting go therefore her kinetic energy must also be the same. However, her kinetic energy will increase from this initial value after she lets go since she is now being accelerated toward the ground by gravity. When she hits the ground she will experience a vertical force upward which will decelerate her vertical motion. Since there is no friction, however, there is no force to change her horizontal velocity so her kinetic energy will decrease back to its original value.
Let's say you've decided to try bungee jumping. You are standing at the top of a tall platform with one end of a bungee cord tied to you and the other end tied to the platform. When you step off the platform you don't push off or jump up so you begin falling straight down. After falling a certain distance the bungee cord becomes taught and begins to stretch. Finally, your fall begins to slow and you come to a complete stop just above the ground before being sent back up by the acceleration provided by the stretched bungee cord.
5. Assuming that the bungee cord is perfectly elastic (no energy is converted into thermal energy) and neglecting air resistance how high will you go after the first bounce? Describe why using energy conservation.
Answer/Why: You will bounce up to the same height from which you started. Initially you have a large gravitational potential energy; you are high above the ground. After stepping off of the platform this gravitational potential energy begins to convert into kinetic energy. When the bungee cord becomes taught energy begins flowing into the cord as elastic potential energy. At the bottom of your fall all of the energy is stored as elastic potential energy. Since we have assumed that no energy is lost to things like heat or air resistance, the flow of energy is exactly reversed on the way back up and you end up at the height of the platform with all of your energy again stored as gravitational potential energy.
6. If you use two identical bungee cords (both with one end attached to you and the other end attached to the platform) when you stop briefly at the bottom will you be lower or higher than with one bungee? Why?
Answer: You will stop at a position that is higher relative to the ground than with one bungee cord. As the two bungee cords stretch together they will provide twice the force of a single bungee cord stopping you earlier.
Why: Since the force increases twice as fast as the pair of bungee cords is stretched during your fall it will take less time to stop you and so you will not fall as far.
7. In a real bungee jump some energy will be wasted in heating up the bungee cord and in moving the air as it resists your motion. You will eventually stop bouncing and come to rest in a stable equilibrium. At this equilibrium position how does the length of the bungee cord compare to its length when you were standing on the platform?
Answer: At the stable equilibrium (when you are hanging motionless) the bungee will be longer than when you were standing on the platform.
Why: On the platform the bungee cord is not being pulled on or stretched, but when you are hanging from it, it must stretch out enough to provide the force required to balance the gravitational force pulling you down, your weight.
8. At this equilibrium position what force is the bungee cord exerting on you? What force do you exert on the bungee cord?
Answer: The force exerted on you by the bungee cord is exactly equal to your weight and points directly up. The force that you exert on the bungee cord is also exactly equal to your weight, but points directly down.
Why: At the equilibrium position you are no longer accelerating so the force exerted on you by the bungee cord must exactly balance the force with which gravity is pulling you down, your weight. Newton's third law states that if the bungee cord is pulling on you with a certain force then you must also be pulling on the bungee with a force of the same size, but the opposite direction.