Orbital Mechanics:

A small space probe is put into circular orbit about a newly discovered moon of Saturn. The moon’s radius is known to be 550 km. If the probe orbits at a height of 1500 km above the moon’s surface and takes 2.0 earth days to make one orbit, determine the moon’s mass. Consider the total radial distance required to solve this problem. Be sure to perform all the necessary dimensional conversions to mks units.

A boy on a carnival bungee ride is released from rest at the bottom of the ride and oscillates up and down with an amplitude of 8 meters and a period of 4s. How long after he is released does he first get to a height 10m above the place where he started? [Hint: Choose the top of the poles as y=0, and measure positive downward.]

A) For hydrogen, find the frequency of light emitted in the transition from the 130th orbit to the 122th orbit.


B) For hydrogen, find the frequency of light absorbed in the transition from the 168th orbit to the 173th orbit.

A bullet with mass 25g and initial horizontal velocity 320m/s strikes a block of mass 2kg that rests on a frictionless surface and is attached to one end of a spring. The bullet becomes embedded in the block. The other end of the spring is attached to the wall. The impact compresss the spring a maximum distance of 25cm. After the impact, the block moves in simple harmonic motion.

1. What is the frequency of the oscillation?

2. Sketch graphs for the block’s position, velocity, and acceleration as a func- tion of time (x vs. t, v vs. t, a vs. t). Check to make sure your graphs are consistent.

3. Write an description of what is happening to the energy in this system from the time the bullet is fired until the block returns to its starting position for the first time. Meaning, tell a story about how the energy changes form or moves from one object to another.

4. Prove that if there are two operators A, ˆ Bˆ, such that their commutator [A, ˆ Bˆ] = λ where
λ is a constant. Show that (6 points)

exp[μ(Aˆ + Bˆ)] = exp(μAˆ) exp(μBˆ) exp(−μ
2λ/2)

where given an operator Xˆ; exp(tXˆ) = 1 + tXˆ +
1
2
t
2Xˆ 2 + ... +
1
n!
t
nXˆ n + ... (the usual

exponential series implemented for operators).

1. You release a disk (momentum of inertia Idisk=(1/2)mr2 ) with mass m = 100 g and radius r = 10 cm from height 15 cm on a ramp with angle 15°, write down the energy conservation equation for the object at the bottom of the ramp.

2. Use the energy conservation equation you wrote down from step 1, solve for the velocity of the disk at the bottom of the ramp.

3. Does your answer of the final velocity of the disk depend on its mass or radius? Explain your answer.

4. If the disk you rolled down the ramp were twice as heavy (i.e., if it had twice the mass), how would this affect your results?

5. If the disk you rolled down the ramp were twice as large (i.e., if it had twice the radius), how would this affect your results?

6. If you rolled any disk down a ramp 15 cm high, what is its speed at the bottom?

7. If you rolled any ring down a ramp 15 cm high, what is its speed at the bottom? Note that for the ring, its momentum of inertia Iring = mr2 .

8. If you rolled a sphere, disk, and ring at the same time, in what order do they reach the bottom? Assume the height is 15 cm.

9. If you dropped a sphere, disk, and ring at the same time, in what order do they hit the ground? Assume the height is 15 cm.

Derive T=MBsin(θ)=NIABsin(θ)

A dockworker loading crates on a ship finds that a 34-kg crate, initially at rest on a horizontal surface, requires a 73-N horizontal force to set it in motion. However, after the crate is in motion, a horizontal force of 50 N is required to keep it moving with a constant speed. Find the coefficients of static and kinetic friction between crate and floor

A bullet weighing 0.1 lb is fired vertically downward from a stationary helicopter with a muzzle velocity of 1200 ft/sec. The air resistance (in lbs) is numerically equal to 16^-5v², where v is the velocity of the bullet in ft/sec.

a)If the helicopter is 3,000 ft high, how long does it take for the bullet to hit the ground?

Two friends are playing golf. The first friend hits a golf ball on level ground with an initial speed of 43.5 m/s at an angle of 32.0° above the horizontal

. (a) Assuming that the ball lands at the same height from which it was hit, how far away from the golfer, in meters, does it land? Ignore air resistance.

(b) The second friend hits his golf ball with the same initial speed as the first, but the initial velocity of the ball makes an angle with horizontal that is greater than 45.0°. The second ball, however, travels the same horizontal distance as the first, and it too lands at the same height from which it was hit. What was the angle in degrees above horizontal of the initial velocity of this second golf ball? Ignore air resistance. ° above the horizontal

On a frictionless table, two stationary balls are placed along the y-axis with their contact being centered on the origin of the XY-plane. A third ball moves along the x-axis toward the pair with an initial velocity V. What will be the velocity (magnitude and angle) of all three balls after the collision? Assume that the three balls are physically identical and that the collisions are perfectly elastic. Show all work and explain the results.

Suppose that a parallel-plate capacitor has circular plates with radius R = 28 mm and a plate separation of 5.5 mm. Suppose also that a sinusoidal potential difference with a maximum value of 140 V and a frequency of 77 Hz is applied across the plates; that is,

V = (140 V) sin[2π(77 Hz)t].

Find B_max(R), the maximum value of the induced magnetic field that occurs at r = R.

You throw a baseball of mass 178 g horizontally from the top of a very tall tower with an initial speed of 11.5 m/s. Determine the magnitude of the vertical component of the ball's velocity 3.2 s after it is thrown. Air resistance can be neglected.