### Chapter 10

Conceptual Questions: 2, 6, 11, 17, 18

2. A grandfather clock is running too fast. To fix it, what should you do?

The problem with a grandfather clock that runs too fast is that first you have to catch the darn thing. Running grandfather clocks are especially rare (since they don't have legs), but once they get going they're really difficult to run down.

Oh. Ooops.

"Running too fast" means, of course, that the period of the pendulum's swing is too short. You want to increase the period of oscillation (also known as decreasing the frequency), so you increase the length.

6.Explain how the period of a mass-spring system can be independent of amplitude, even though the distance traveled during each cycle is proportional ot the amplitude.

This is one of my favorite relationships in physics. As you increase the distance that the mass needs to travel, you also put more energy into the system. So, it moves faster (on average) as a result. It turns out that the amount it moves faster is exactly the right amount to compensate for the extra distance it has to go. How does it know to do this, just right, no matter what the amplitude is? I don't know. It just does, which makes the physical system very simple and predictable.

11. A pilot is performing vertical loop-the-loops over the ocean at noon. The plane speeds up as it approachhes the bottom of the circular loopand slows down as it approaches the top of the loop. An observer in a helicopter is watching the shadow of the plane on the surface of the water. Does the shadow exhibit SHM? Explain.

And also, there's a flying saucer and a shark that jumps out of the water and gobbles up the plane. And then the flying saucer scoops up the shark (with plane ingested) and flies away, leaving the helipcoter observer to sit and think, quietly, "hmmmmm, I wonder if that motion was SHM?"

Stupid physicists. Where do they come up with these situations?

As seen from above, the oscillation of the plane will be going back and forth with the same amplitude each time, but the speed at which it travels changes with every half cycle. That is, as it's going one way (at the bottom of the loop) it's traveling quickly; but while going the opposite direction (at the top of the loop) it's traveling slowly. You can imagine that this messes up the symmetry of the graphs of position versus time, velocity versus time, etc. Every other half oscillation would have a different portion of the period. Or, said another way, it wouldn't look like the force is directly proportional to displacement. So this would not mimic SHM.

17. The period of oscillation of a simple pendulum does not depend on the mass of the bob. By contrast, the period of a mass-spring system does depend on mass. Explain this apparent contradiction.

The reason the simple pendulum has no dependence on mass is because the mass gets to "count" for two different things. (The same thing happens in freefall motion, where all things of all weights fall at the same rate.) Mass counts for inertia, or the "m" in "F=ma". That means the resistance to changes in motion is directly proportional to the mass. However, the weight (a force) on an object is also proportional to mass. Since the mass factors into both the cause of changing motion and the resistance to changing motion, it cancels out.

For a mass-spring system, the mass still affects the inertia, but it does not cause the force. The spring (and its spring constant) is fully responsible for force. So mass only impacts the resistance to accelerations, and you notice that the more massive the object the slower it wiggles back and forth.

18. A mass connected to an ideal spring is oscillation without friction on a horizontal surface. Sketch graphs of the kinetic energy, potential energy, and total energy as functions of time for one complete cycle.

[I did this in the problem set. See number 53.]