### Additional Problems

A marble is rolling across the floor at a speed of 7.0 m/s when it starts up a plane inclined at $30\text{\xb0}$ to the horizontal. (a) How far along the plane does the marble travel before coming to a rest? (b) How much time elapses while the marble moves up the plane?

Repeat the preceding problem replacing the marble with a hollow sphere. Explain the new results.

The mass of a hoop of radius 1.0 m is 6.0 kg. It rolls across a horizontal surface with a speed of 10.0 m/s. (a) How much work is required to stop the hoop? (b) If the hoop starts up a surface at $30\text{\xb0}$ to the horizontal with a speed of 10.0 m/s, how far along the incline will it travel before stopping and rolling back down?

Repeat the preceding problem for a hollow sphere of the same radius and mass and initial speed. Explain the differences in the results.

A particle has mass 0.5 kg and is traveling along the line $x=5.0\phantom{\rule{0.2em}{0ex}}\text{m}$ at 2.0 m/s in the positive *y*-direction. What is the particle’s angular momentum about the origin?

A 4.0-kg particle moves in a circle of radius 2.0 m. The angular momentum of the particle varies in time according to $l=5.0{t}^{2}.$ (a) What is the torque on the particle about the center of the circle at $t=3.4\phantom{\rule{0.2em}{0ex}}\text{s}$? (b) What is the angular velocity of the particle at $t=3.4\phantom{\rule{0.2em}{0ex}}\text{s}$?

A proton is accelerated in a cyclotron to $5.0\phantom{\rule{0.2em}{0ex}}\text{\xd7}\phantom{\rule{0.2em}{0ex}}{10}^{6}\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}\text{s}$ in 0.01 s. The proton follows a circular path. If the radius of the cyclotron is 0.5 km, (a) What is the angular momentum of the proton about the center at its maximum speed? (b) What is the torque on the proton about the center as it accelerates to maximum speed?

(a) What is the angular momentum of the Moon in its orbit around Earth? (b) How does this angular momentum compare with the angular momentum of the Moon on its axis? Remember that the Moon keeps one side toward Earth at all times.

A DVD is rotating at 500 rpm. What is the angular momentum of the DVD if has a radius of 6.0 cm and mass 20.0 g?

A potter’s disk spins from rest up to 10 rev/s in 15 s. The disk has a mass 3.0 kg and radius 30.0 cm. What is the angular momentum of the disk at $t=5\phantom{\rule{0.2em}{0ex}}\text{s,}\phantom{\rule{0.2em}{0ex}}t=1\text{0 s}$?

Suppose you start an antique car by exerting a force of 300 N on its crank for 0.250 s. What is the angular momentum given to the engine if the handle of the crank is 0.300 m from the pivot and the force is exerted to create maximum torque the entire time?

A solid cylinder of mass 2.0 kg and radius 20 cm is rotating counterclockwise around a vertical axis through its center at 600 rev/min. A second solid cylinder of the same mass and radius is rotating clockwise around the same vertical axis at 900 rev/min. If the cylinders couple so that they rotate about the same vertical axis, what is the angular velocity of the combination?

A boy stands at the center of a platform that is rotating without friction at 1.0 rev/s. The boy holds weights as far from his body as possible. At this position the total moment of inertia of the boy, platform, and weights is $5.0\phantom{\rule{0.2em}{0ex}}\text{kg}\xb7{\text{m}}^{2}.$ The boy draws the weights in close to his body, thereby decreasing the total moment of inertia to $1.5\phantom{\rule{0.2em}{0ex}}\text{kg}\xb7{\text{m}}^{2}.$ (a) What is the final angular velocity of the platform? (b) By how much does the rotational kinetic energy increase?

Eight children, each of mass 40 kg, climb on a small merry-go-round. They position themselves evenly on the outer edge and join hands. The merry-go-round has a radius of 4.0 m and a moment of inertia $1000.0\phantom{\rule{0.2em}{0ex}}\text{kg}\xb7{\text{m}}^{2}$. After the merry-go-round is given an angular velocity of 6.0 rev/min, the children walk inward and stop when they are 0.75 m from the axis of rotation. What is the new angular velocity of the merry-go-round? Assume there is negligible frictional torque on the structure.

A thin meter stick of mass 150 g rotates around an axis perpendicular to the stick’s long axis at an angular velocity of 240 rev/min. What is the angular momentum of the stick if the rotation axis (a) passes through the center of the stick? (b) Passes through one end of the stick?

A satellite in the shape of a sphere of mass 20,000 kg and radius 5.0 m is spinning about an axis through its center of mass. It has a rotation rate of 8.0 rev/s. Two antennas deploy in the plane of rotation extending from the center of mass of the satellite. Each antenna can be approximated as a rod has mass 200.0 kg and length 7.0 m. What is the new rotation rate of the satellite?

A top has moment of inertia $3.2\phantom{\rule{0.2em}{0ex}}\text{\xd7}\phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}\phantom{\rule{0.2em}{0ex}}\text{kg}\xb7{\text{m}}^{2}$ and radius 4.0 cm from the center of mass to the pivot point. If it spins at 20.0 rev/s and is precessing, how many revolutions does it precess in 10.0 s?