#### Practice Problems for Final Exam

1. Professor F floats H = 200 m above the ground in his hot-air balloon. He spots his enemy Fresh-Man sleeping below on the ground. Cackling evilly, he drops a ringing alarm clock towards our hero. The clock rings at f = 100 Hz, but Fresh-Man's ears don't pick up sounds below f2 = 120 Hz.

1. Will our hero hear the clock before it lands on him?
2. If so, how much time will he have to wake up and roll out of the way?

The detailed solution:

2. A circular disk of radius R is made of material which is very dense around the edges, but not very dense near the center. The surface density (mass per unit area) at a distance r from the center is described by the equation
```                                         (       r  )
surface mass density =   K * (  1 + --- )
(       R  )
```
where K is some constant with units of kilograms per square meter.

1. What is the mass of this disk, in terms of K and R?
2. What is the moment of inertia of this disk around its center?
3. The disk is put on a ramp which is tilted at angle theta above the horizontal. The starting point of the disk is a vertical height H above the bottom of the ramp. The disk is held motionless, then released. It starts to roll without slipping.

4. How long will it take the disk to roll down to the bottom of the ramp?

The detailed solution:

3. The laser lab is a circular room R = 6 m in radius. A laser is mounted at the North end of the room, pointing towards the center. At the center is a long, thin mirror, polished on both sides, of mass m = 0.6 kg and length L = 40 cm. The mirror is mounted on a rod which can be rotated by a motor.

Initially, the mirror is oriented East-West, so that the laser beam bounces straight back at the laser.

At time t = 0, Polly turns on the motor, which applies a constant torque 0.03 N*m to the mirror. The mirror begins to spin in a counter-clockwise direction.

1. What is the orientation of the mirror at t = 5 seconds?
2. At this moment, where does the reflected laser beam strike the wall of the lab? Express your answer as the distance Polly would measure along the surface of the wall, starting at the laser and progressing clockwise.

The detailed solution:

4. The amusement park builds a big wave tank. The waves are created by a simple machine: a big panel which is pushed forward by 0.8 m, the back again, repeatedly, every 1.6 s. The wave crests move through the tank with a speed of 60 m/s. One particular crest appears at x = 5 m and t = 2 s.

1. Write an equation which expresses the height of the surface of the water as a function of position and time:
```                  y(x,t) = sin(kx - wt + phi)
```
Provide values for k, w, phi.
2. The foreman realizes that he could increase the height of the waves by making the length of the tank a particular length. What length will do the trick?

The detailed solution: