#### Practice Problems for Test 3

1. Joe has a thin, springy, elastic cord of mass M = 12 g and force constant k = 5 N/m. He pulls it with tension T = 25 N so that the cord has a length of L = 3 m.

Joe plucks one end of the cord. A disturbance travels down to the far end of the cord, reflects and comes back to Joe.

1. What is the speed of the wave as it travels down the string?
2. How long does it take the wave to make the round trip?
3. Now Joe increases the tension to 30 N, which pulls the cord to a new total length.

4. What is the new length of the cord?
5. Joe plucks his end again.

6. How long does it take the wave to make the round trip now?
2. The detailed solution:

3. Rita has an oscillator which produces a pure tone of frequency 400 Hz. She hangs it from a long string of length L, pulls it an angle 0.1 radian from vertical, and releases it. The oscillator swings back and forth. As it comes towards her, the frequency goes up -- as it moves away, the frequency goes down. The maximum frequency Rita hears is 401 Hz.

1. How long is the string?
4. The detailed solution:

5. Bill is walking down the street when he sees an enormous ice cube. It looks like a PERFECT cube, with perfectly pure ice. Bill has a red laser of wavelength 627 nm in the air. He aims it up at an angle 30 degrees above the horizontal and shines it at the cube. He sees a little red spot where the laser enters the cube. Looking further up, he sees a second, fainter red spot, exactly 10 m above the first spot. "Hmmm", thinks Bill, "some of the light must enter the cube, bounce off the back wall, then come out the front face up there."

Bill knows that the index of refraction of air is n(air) = 1.00029, and the index of refraction of ice is n(ice) = 1.31.

1. At what angle from the normal does the laser light travel through the cube after entering the front face?
2. How long is each side of the ice cube?
3. Now, pretend for a moment that all calculations can be carried out to 10 digits. In real life, things are never this perfect.

4. If Bill could compare the two beams of laser light which head out from the cube -- the lower beam, which reflects off the front face, and the upper beam, which comes out after travelling inside the ice -- would they be (roughly) in phase or out of phase?

The detailed solution: