Q: A Formula One car has tires of radius 33 cm and travels at an average speed of 83 m/s for 1.5 hours. What is the angular displacement of a tire during the race?

$\Delta \theta = \frac{v_{\mathrm{CM}}}{R} \Delta t = \frac{83}{0.33} \times 5400 \approx 1.36 \times 10^6$ radians

Q: A 32.0 kg thin hoop with radius 1.20 m rotates at 28.0 rad/s. How much work is required to stop it, and what is the average power if it is stopped in 15.0 s?

Work: $W = \frac{1}{2} I \omega^2 = \frac{1}{2} (32.0 \times 1.20^2) \times 28.0^2 \approx 20,700$ J; Power: $P = \frac{W}{t} \approx 1,380$ W

Q: Which law must be applied to analyze a system that is both translating and rotating?

Combine $\Sigma F = ma$ and $\Sigma \tau = I \alpha$

Question 1

A 2 kg bucket is attached to a massless string wrapped around a 4 cm diameter cylinder (assume no friction). If the bucket is released from rest and falls 2 meters, how long does it take to reach the bottom? (Assume $M$ is the mass of the cylinder and $m$ is the mass of the bucket. Use $g = 9.8 \ \mathrm{m/s^2}$.)

Accepted Answer

Use $a = \frac{m}{m + \frac{1}{2}M}g$ and $t = \sqrt{\frac{2h}{a}}$; with $M = 0$, $t \approx 0.64$ s

Question 2

Which of the following correctly expresses the rolling constraint for a wheel rolling without slipping?

Accepted Answer

$v_{\mathrm{CM}} = \omega R$

Question 3

A Formula One car has tires of radius 33 cm and travels at an average speed of 83 m/s for 1.5 hours. What is the angular displacement of a tire during the race?

Accepted Answer

$\Delta 	heta = \frac{v_{\mathrm{CM}}}{R} \Delta t = \frac{83}{0.33} 	imes 5400 \approx 1.36 	imes 10^6$ radians

Question 4

A 32.0 kg thin hoop with radius 1.20 m rotates at 28.0 rad/s. How much work is required to stop it, and what is the average power if it is stopped in 15.0 s?

Accepted Answer

Work: $W = \frac{1}{2} I \omega^2 = \frac{1}{2} (32.0 	imes 1.20^2) 	imes 28.0^2 \approx 20,700$ J; Power: $P = \frac{W}{t} \approx 1,380$ W

Question 5

A neutron star forms from a stellar core of radius $R_1 = 7 	imes 10^5$ km, mass $M$, rotating once every 10 days. After collapse, its radius is $R_2 = 100$ km. Assuming a solid sphere, what is its new rotation period?

Accepted Answer

Use conservation of angular momentum: $I_1 \omega_1 = I_2 \omega_2$; $T_2 = T_1 \left( \frac{R_2}{R_1} ight)^2 \approx 10 	imes \left( \frac{100}{7 	imes 10^5} ight)^2$ days $\approx 2.04 	imes 10^{-5}$ days

Question 6

A 20.0-cm diameter, 1.0-kg solid disk rotates at 200 rpm. A 20.0-cm diameter, 1.0-kg circular loop is dropped onto it and rides with it. What is the final angular velocity of the system?

Accepted Answer

Conservation of angular momentum: $I_{\mathrm{disk}} \omega_i = (I_{\mathrm{disk}} + I_{\mathrm{loop}}) \omega_f$; $\omega_f = \frac{I_{\mathrm{disk}}}{I_{\mathrm{disk}} + I_{\mathrm{loop}}} \omega_i = \frac{0.01}{0.02} 	imes 200 = 100$ rpm

Question 7

Which of the following is the correct expression for the moment of inertia of a thin hoop of mass $M$ and radius $R$ about its central axis?

Accepted Answer

$I = MR^2$

Question 8

A solid sphere of mass $M$ and radius $R$ rolls down a ramp without slipping. What is its speed at the bottom if it starts from height $h$?

Accepted Answer

$v_{\mathrm{CM}} = \sqrt{\frac{2gh}{1 + \frac{2}{5}}}$

Question 9

Which statement about rolling without slipping is correct?

Accepted Answer

The point of contact with the ground has zero velocity relative to the ground.

Question 10

If a wheel is rolling without slipping, what is the velocity of the top point of the wheel relative to the ground?

Accepted Answer

$v_{\mathrm{top}} = 2 v_{\mathrm{CM}}$

Question 11

Which of the following is the correct expression for the total kinetic energy of a rolling object?

Accepted Answer

$KE = \frac{1}{2} m v_{\mathrm{CM}}^2 + \frac{1}{2} I \omega^2$

Question 12

Which law must be applied to analyze a system that is both translating and rotating?

Accepted Answer

Combine $\Sigma F = ma$ and $\Sigma 	au = I \alpha$

Question 13

A solid disk and a thin hoop of equal mass and radius roll down the same ramp from rest. Which reaches the bottom first?

Accepted Answer

The solid disk, because it has a smaller moment of inertia relative to its mass.

Question 14

Which of the following is true about angular momentum in the absence of external torque?

Accepted Answer

Angular momentum is conserved: $I_1 \omega_1 = I_2 \omega_2$

Question 15

A wheel of mass $M$ and radius $R$ is rotating with angular velocity $\omega$. What is its angular momentum?

Accepted Answer

$L = I \omega = MR^2 \omega$

Question 16

A solid cylinder of mass $M$ and radius $R$ rotates about its axis. What is its moment of inertia?

Accepted Answer

$I = \frac{1}{2} MR^2$

Question 17

If a rolling object is subject to static friction, what is the role of friction in rolling without slipping?

Accepted Answer

Static friction provides the torque necessary for rolling without slipping.

Question 18

Which of the following is the correct relationship for the acceleration of a bucket falling while attached to a rotating cylinder (mass $M$, bucket mass $m$)?

Accepted Answer

$a = \frac{m}{m + \frac{1}{2}M}g$

Rotation III: Rolling Motion, Rotational Kinetics, and Angular Momentum

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