Problem Set #2
Goal of Problem Set #2: This assignment is meant to help you understand:
  1. three conserved quantities—energy, momentum, and angular momentum,
  2. how those conserved quantities are transferred between objects,
  3. the relationships between kinetic and potential energies,
  4. equilibrium in general and stable equilibrium in particular,
  5. restoring forces in general and spring forces in particular.
You’ve had enough reading for now, so you decide to head south with some friends for the fall reading holiday. Way south... to the South American rain forest and the Amazon River. At this very moment, you are perched motionless half-way across a long rope bridge. Far below you is a deep river and a distant waterfall. You're terrified, partly by the height and partly by the bridge's tendency to move. Also, you're wearing a dry-clean-only shirt and it water spots easily. You don't want to be on the bridge any longer than necessary, so let's get started with the questions.
1. Though it has no true springs in it, the bridge behaves like one. In effect, you are standing motionless on a huge spring. Briefly explain why you are now at a stable equilibrium.
2. Before you walked to the middle of the bridge, its motionless surface was 5 feet higher than it is now. Your friend, who weighs the same amount as you, now walks slowly onto the bridge from one end and joins you in the middle. The bridge is again motionless and the two of you are at a stable equilibrium. Has the bridge's surface moved up or down and, if so, by how much?
Your friend is always kind and well-meaning, but can't resist the urge to jump up and down. After a couple of strategically timed hops, the two of you find yourselves bouncing wildly up and down. The bouncing is steady and prolonged, so let's imagine that each bounce is exactly like the one before it. You are gripping the rope railings so tightly that you are leaving fingerprints in them. Your terror and vague concerns about consequences are all that protect your friend from immediate strangulation.
3. Because you are holding the railings so tightly, you are bouncing with the bridge and never leave its surface. At what moment(s) during a single complete bounce (up and down),
(A) are you traveling upward fastest,
(B) are you traveling downward fastest,
(C) are you accelerating upward fastest,
(D) are you accelerating downward fastest, and
(E) do you have the greatest kinetic energy?
4. The bouncing gradually diminishes and you are again motionless at the middle of the bridge. You release your grip on the rope railings just as your cheerful friend slaps you hard on the back. Use the concept of an impulse to prove that this slap transfers horizontal momentum from your friend to you.
5. Your new momentum carries you off the side of the bridge and you suddenly having nothing under your feet. Down you go! As you fall, how do the following quantities change (or not change):
(A) your gravitational potential energy,
(B) your kinetic energy, and
(C) your total energy?
(Assume no air resistance and that you don’t touch anything as you fall.)
6. As you fall feet-first, you notice that you are spinning about a vertical axis like a toy top. You try to stop spinning, to get a better view of what lies below you, but nothing you do seems to stop your spin. Use the concept of angular momentum to explain briefly why you can’t stop spinning but why you can slow your rate of rotation by extending your arms out horizontally.
7. Below you are two possible targets: deep water and a wide, flat rock. Hitting either one will stop your downward motion, but the water will stop you more comfortably than the rock. Use the concepts of momentum and impulse to explain why it would feel better to hit the water than to hit the rock.
8. Luckily, you hit the water and float safely back to the surface. After swimming to shore, you scramble up the river bank to one end of the bridge. Together with some other members of your group, you grab hold of that end of the bridge and pull on it horizontally as hard as you can. The bridge suddenly pulls taut: it stops drooping in the middle and becomes almost perfectly straight. As the bridge rises toward horizontal, your friend, who was still in the middle of the bridge, is tossed about 20 feet into the air above the bridge. Where did the energy that lifted your friend upward come from? Use the concept of work to prove that energy was transferred first from you to the bridge and second from the bridge to your friend?
Your friend lands safely back on the bridge and walks over to join you at the end. You both realize the folly of your ways and let bygones be bygones. Friends once more, you all head off into the sunset...