Problem Set #10
Goal of Problem Set #10: This assignment is meant to help you understand:
  1. the transformations between materials phases,
  2. the nature of harmonic oscillation.
As manager for an overnight delivery service, you often find yourself flying from place to place with the cargo and urging your staff to work harder and faster because the clock is everything in your life (citation: Tom Hanks, with apologies). One day, your cargo plane crashes into the South Pacific and you find yourself alone on an island with a bunch of overnight packages. One package contains a bowling ball (why would anyone send a bowling ball by overnight express?). You paint a face on the bowling ball and name it "Newton". Actually, her full name is "Ms. Newton" because she's pink. Her first name is "Fig", of course. You settle in for a long wait, knowing that rescue will take time.

1. You soon begin to miss your favorite drink: iced tea. There are plenty of tea plants around and you succeeded in making fire several days back. You have learned to collect rainwater in a now-empty pretzel can and heat that water over the fire. When it begins to boil, you toss in some tea and let it steep. It's a cool night and while you wait for the tea, you warm your hands in the vapor rising from the hot tea water. Your hands quickly become damp, but they warm up nicely. Why is this vapor so effective at warming your hands and why does it make your hands damp?

2. It truly is a cold night and another pretzel can containing rainwater is cooling down toward 0 °C. When it gets to 0 °C, its temperature stops dropping and ice begins to form. Throughout that ice formation, the water's temperature stays steady at 0 °C. Is heat still flowing out of the pretzel can as the ice is forming?

3. Success. You have your ice and you have your hot tea. You mix them together. After a few minutes, you have liquid tea with ice floating around in it. What is the temperature of that tea?

4. When you wake up in the morning, the ground is much colder than 0 °C but the sea and the air are somewhat warmer than 0 °C. The ground is covered with frost. Why did the frost form?

Your favorite time to drink tea is at tea-time: 4:00pm. But you can't tell when it's 4:00pm because none of the packages contains a clock. That's pretty astonishing since just about everything these days has a clock in it. So you use a vine to hang Ms. Newton from a palm tree and you give her a push. She swings back and forth steadily and when she completes her 10,280th swing, starting with sunrise, it's tea-time! Yes, I know that it's boring to count to 10,280, but what else do you have to do? Just be glad you didn't drink too much tea before you started counting.

5. Ms. Newton has an equilibrium position located directly below the tree. When you displace her from the equilibrium position, she experiences a restoring force (a push back toward equilibrium). How does the strength of that restoring force vary as her distance from equilibrium increases?

6. As Ms. Newton swings back and forth, does the period of that swing (the time required for her to complete one full round trip swing) depend significantly on the size of the swing (big or small)? If so, which type of swing takes the least time?

7. To keep Ms. Newton swinging, you occasionally have to give her a push. Do you push her forward as she comes toward you or as she moves away from you? Why?

8. To reduce the number of swings you must count before tea-time, you change Ms. Newton's arrangement slightly. What do you change to make each swing take more time?

Epilogue: A sailor on a passing boat spies Ms. Newton swinging back and forth and is instantly hypnotized by her. When he wakes up, he asks the captain to steer the boat toward your island. When you come into view, a second sailor of your favorite persuasion descends from the boat, rescues you, and sweeps you off your feet. In appreciation (and to avoid lawsuits), your company gives you a bazillion dollars and you and your new companion sail off into the sunset while Ms. Newton completes swing number 15,281, 15,282, 15,283...