1. the transformations between materials phases,
2. the nature of harmonic oscillation.

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?

Answer: The water vapor in the rising gas condenses on your hands, releasing chemical potential energy as heat. Your hands are warmed as they become damp with the condensing water.

Why: At high temperature, the equilibrium between liquid water and gaseous water is achieved when there is a dense vapor of gaseous water. But at lower temperatures, the gaseous vapor can't be that dense and still be in equilibrium with the liquid. The gas condenses to liquid, releasing chemical potential energy and warming your hands nicely.

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?

Answer: Yes.

Why: To convert water at 0 °C into ice at 0 °C, something has to remove thermal energy. The water molecules form tighter bonds as the solid form develops and the chemical potential energy released during that bond formation must be drawn away as heat.

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?

Answer: 0 °C.

Why: Liquid water and solid ice can only coexist at one temperature: the freezing (or equivalently melting) temperature.

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?

Answer: The density of water molecules in the warm, moist air is high and so they land on the ground frequently. The ground is cold, so water molecules there rarely take off. There is an imbalance and more water molecules land than take off, forming frost.

Why: Frost forms when water vapor condenses directly to ice. That happens here because there is too dense a vapor of water molecules to remain in equilibrium with the cold frost. The gaseous water phase shrinks while the solid water phase grows.

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?

Answer: The restoring force increases in proportion to her distance from equilibrium.

Why: The force on a pendulum is almost exactly proportional to its displacement from equilibrium and Ms. Newton is now a pendulum.

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?

Answer: The period does not depend significantly on the size of the swing.

Why: Since the restoring force acting on Ms. Newton is proportional to her displacement from equilibrium, she is a harmonic oscillator. Harmonic oscillators complete their motions with periods that do not depend on amplitude (size of the motion).

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?

Answer: Push her as she moves away, because then you do work on her and she gains energy.

Why: To add energy to her motion, you must do work on her. You must push her and she must move in the direction of your force on her.

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?

Answer: Lengthen the vine.

Why: Lengthening the vine weakens the restoring force on Ms. Newton and prolongs each swing.

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...