Q: A wheel of diameter 68.0 cm slows uniformly from $$v_0 = 8.40$$\ \mathrm{m/s}$$ to rest over a distance of 115 m. What is the angular displacement $$\Delta \theta$$ in radians during this process?

$$\Delta \theta = \dfrac{115}{0.34}$$

Q: Which kinematic equation for rotational motion under constant angular acceleration is analogous to $$v^2$$ = $$v_0^2 + 2a(x$$ - $$x_0$$)$$?

$$\$$omega^2$$ = \$$omega_0$$^2 + 2\alpha (\theta - \$$theta_0$$)$$

Q: A pulley of mass $$m = 4.00\ \mathrm{kg}$$ and radius $$r = 0.33\ \mathrm{m}$$ accelerates uniformly from rest to $$\omega = 30.0\ \mathrm{rad/s} in$$ $$3.00\ \mathrm{s}$$ under a net torque of $$\tau = 1.10\ \mathrm{N\cdot m}. $$What is the moment of inertia $$I of $$the pulley?

$$I = \dfrac{\tau}{\alpha}$$, where $$\alpha = \dfrac{\omega}{t}$$

Q: If a bucket of mass $$m = 1.53\ \mathrm{kg}$$ hangs from a cord wrapped around a pulley with $$I = 0.385\ \mathrm{kg\cdot m^2$}$$ and $$r = 0.33\ \mathrm{m}$$, and the frictional torque is $$\tau_f = 1.10\ \mathrm{N\cdot m}$$, what is the equation for the angular acceleration $$\alpha of $$the pulley?

$$\alpha = \dfrac{m g r - \tau_f}{I + m r^2$}$$

Question 1

Which of the following best describes the difference between translational and rotational motion for a rigid body?

Accepted Answer

Translational motion describes the movement of the center of mass from one point to another, while rotational motion describes the rotation of the object about an axis with the center of mass remaining stationary.

Question 2

A wheel of diameter 68.0 cm slows uniformly from $$v_0 = 8.40$$\ \mathrm{m/s}$$ to rest over a distance of 115 m. What is the angular displacement $$\Delta 	heta$$ in radians during this process?

Accepted Answer

$$\Delta 	heta = \dfrac{115}{0.34}$$

Question 3

Which equation correctly relates angular velocity $$\omega$$ and linear velocity $$v$$ for a point at radius $$r$$ from the axis of rotation?

Accepted Answer

$$v = \omega r$$

Question 4

A force $$F is $$applied at a distance $$r$$ from the pivot at an angle $$	heta to $$the lever arm. What is the torque $$	au$$ produced?

Accepted Answer

$$	au = Fr \sin 	heta$$

Question 5

Which of the following statements about the moment of inertia $$I is $$correct?

Accepted Answer

The moment of inertia depends on both the mass distribution and the axis of rotation.

Question 6

A dumbbell consists of masses 5.0 kg and 7.0 kg mounted 4.0 m apart on a light rod. What is the moment of inertia about an axis halfway between the weights?

Accepted Answer

$$I = (5.0)(2.0^2$) + (7.0)(2.0^2$)$$

Question 7

Which kinematic equation for rotational motion under constant angular acceleration is analogous to $$v^2$$ = $$v_0^2 + 2a(x$$ - $$x_0$$)$$?

Accepted Answer

$$\$$omega^2$$ = \$$omega_0$$^2 + 2\alpha (	heta - \$$theta_0$$)$$

Question 8

A pulley of mass $$m = 4.00\ \mathrm{kg}$$ and radius $$r = 0.33\ \mathrm{m}$$ accelerates uniformly from rest to $$\omega = 30.0\ \mathrm{rad/s} in$$ $$3.00\ \mathrm{s}$$ under a net torque of $$	au = 1.10\ \mathrm{N\cdot m}. $$What is the moment of inertia $$I of $$the pulley?

Accepted Answer

$$I = \dfrac{	au}{\alpha}$$, where $$\alpha = \dfrac{\omega}{t}$$

Question 9

If a bucket of mass $$m = 1.53\ \mathrm{kg}$$ hangs from a cord wrapped around a pulley with $$I = 0.385\ \mathrm{kg\cdot m^2$}$$ and $$r = 0.33\ \mathrm{m}$$, and the frictional torque is $$	au_f = 1.10\ \mathrm{N\cdot m}$$, what is the equation for the angular acceleration $$\alpha of $$the pulley?

Accepted Answer

$$\alpha = \dfrac{m g r - 	au_f}{I + m r^2$}$$

Question 10

Which of the following is the correct conversion between degrees and radians?

Accepted Answer

$$360^\circ = 2\pi\ \mathrm{rad}$$

Question 11

Which statement about the parallel axis theorem is correct?

Accepted Answer

It allows calculation of the moment of inertia about any axis parallel to one through the center of mass.

Question 12

A wheel rotates at a constant angular velocity $$\omega. $$Which point on the wheel has the greatest linear speed?

Accepted Answer

The point farthest from the axis of rotation.

Question 13

If the net torque on a rotating object is zero, what can be said about its angular acceleration?

Accepted Answer

The angular acceleration is zero.

Question 14

Which of the following is the correct expression for the moment of inertia of a point mass $$m at a $$distance $$r$$ from the axis of rotation?

Accepted Answer

$$I = m r^2$

Question 15

A wheel slows uniformly from $$v_0 = 8.40$$\ \mathrm{m/s}$$ to rest over a distance of 115 m. What is the angular acceleration $$\alpha$$ if the wheel's radius is 0.34$$\ \mathrm{m}$$?

Accepted Answer

$$\alpha = \dfrac{\$$omega^2$$ - \$$omega_0$$^2}{2 \Delta 	heta}$$

Question 16

Which of the following is NOT a correct analogy between linear and rotational motion?

Accepted Answer

Force $$F$$ corresponds to angular velocity $$\omega.$$

Question 17

A force is applied perpendicular to a lever arm at a distance $$r$$ from the pivot. If the force is doubled and the distance is halved, what happens to the torque?

Accepted Answer

The torque remains the same.

Question 18

Which of the following best explains why objects with mass distributed farther from the axis of rotation have a larger moment of inertia?

Accepted Answer

Because $$I$$ increases with the square of the distance from the axis, making mass farther away contribute more to rotational resistance.

Rotational Kinematics and Dynamics: Study Notes

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