Problem Set #10
Goal of Problem Set #10: This assignment is meant to help you understand:
  1. the characteristics of harmonic oscillators,
  2. the roles of masses and restoring forces in harmonic oscillators,
  3. the behaviors of non-harmonic oscillators,
  4. the flow of energy within resonant systems.
You live only a few hundred yards from the bridge across the Myteedeep Gorge outside of Charlottesville. The ground below the middle of the bridge is just over 1000 feet beneath it and you and your friends decide to open a bungee-jumping company. You buy a variety of elastic cords, each with its own spring-like stiffness. The cords range from easy to stretch to very difficult to stretch. Now all you need is an ironclad insurance policy and you're in business! Of course, you all decide to test everything out first. Since you're the physicist of the bunch, everyone turns to you for insight into how to prepare for the first jump. [Note: neglect air resistance or buoyant effects in this problem.]
1. One friend measures off 1000 feet of elastic cord and ties it between his ankles and the bridge. He is about to jump when you caution him that he has made mistake and that he won't enjoy trip. What is wrong with his using 1000 feet of cord?
2. Another friend is nervous about breaking the cord, despite weighing less than 100 pounds, and she selects the very stiffest of the elastic cords. Although she chooses the appropriate length for that cord, you caution her that she still won't enjoy the trip. What is wrong with her using the stiffest cord?
3. Soon, you have everyone set up with cords of appropriate lengths and stiffnesses. It's time to begin jumping. The first jumper steps off the bridge and plummets toward the ground below. The cord pulls taut and begins to pull upward on him. At the exact moment that the cord first pulls taut, what is your friend's acceleration?
4. Your friend bounces up and down a number of times before coming to rest and you are timing his bounces. During the first two upward bounces, he rises so high that the cord briefly goes slack. After that, the cord remains taut as he travels up and down. You find that the time between those first two upward bounces is longer than between any subsequent bounces. Why?
5. The bounces during which the cord remains taut all take exactly the same amount of time to complete, even though they are of ever decreasing amplitude. Why?
6. Two of your friends decide to jump simultaneously using identical cords. One friend is considerably heavier than the other. They step off the bridge together and begin to bounce up and down. The cords are remaining taut as they go up and down. Do they take equal times to complete their bounces or is one of them bouncing faster than the other? Why?
7. Finally, it's your turn to jump. You step off the bridge and are soon bouncing up and down on the cord attached to your ankles. The cord stays taut as you bounce and you begin to think about the situation. When during your bounces is (A) your kinetic energy at its maximum, (B) at its minimum, and when during your bounces is (C) your total potential energy at its maximum, (D) at its minimum?
8. Your friends are always looking for fun, so they grab the upper end of your bungee cord and begin to yank it upward rhythmically. When they get their timing just right, you begin to bounce more and more wildly up and down. Why was it important that they yank rhythmically at just the right frequency and what frequency did they chose to make you bounce so wildly?