Q: A yo-yo of mass $$m$$ and radius $$R is $$released from rest and descends vertically while spinning. Which equation correctly relates the linear acceleration $$a of $$the center of mass to the tension $$T in $$the string and the moment of inertia $$I$$?

$$mg - T = ma$$ \text{ and } TR = I\alpha$

Q: A disk rotates with constant angular acceleration $$\alpha. If $$its initial angular velocity is $$\$$omega_0$$ and it rotates through an angle $$\theta$$ in time t$, which equation relates these quantities?

$$\theta = \$$omega_0$$ t + \frac{1}{2}\alpha t^2$

Question 1

A mechanic uses a wrench to loosen a bolt. If the mechanic applies a force of 50 N at a distance of 0.3 m from the axis of rotation, but at an angle of 60° to the lever arm, what is the magnitude of the torque produced?

Accepted Answer

$$13.0\ \mathrm{N \cdot m}$$

Question 2

Which of the following statements about torque is correct?

Accepted Answer

Torque is the product of the force and the perpendicular distance from the axis of rotation to the line of action of the force.

Question 3

A solid disk and a hollow ring, both with the same mass and radius, are released from rest at the top of an incline. Which will reach the bottom first and why?

Accepted Answer

The solid disk, because it has a smaller moment of inertia.

Question 4

A wheel of radius $$0.5\ \mathrm{m} is $$rolling without slipping at a speed of $$2\ \mathrm{m/s}. $$What is the angular velocity $$\omega of $$the wheel?

Accepted Answer

$$4\ \mathrm{rad/s}$$

Question 5

A uniform rod of length $$1.2\ \mathrm{m} is $$pivoted at one end. If a force of $$10\ \mathrm{N} is $$applied perpendicular to the rod at its free end, what is the resulting torque about the pivot?

Accepted Answer

$$12.0\ \mathrm{N \cdot m}$$

Question 6

A spinning figure skater pulls in her arms, reducing her moment of inertia to half its original value. If her initial angular velocity was $$\$$omega_0$$, what is her final angular velocity?

Accepted Answer

$$2\$$omega_0$$

Question 7

Which of the following best describes the condition for the conservation of angular momentum in a system?

Accepted Answer

The net external torque on the system is zero.

Question 8

A yo-yo of mass $$m$$ and radius $$R is $$released from rest and descends vertically while spinning. Which equation correctly relates the linear acceleration $$a of $$the center of mass to the tension $$T in $$the string and the moment of inertia $$I$$?

Accepted Answer

$$mg - T = ma$$ 	ext{ and } TR = I\alpha$

Question 9

A force of $$20\ \mathrm{N} is $$applied tangentially to the rim of a solid disk of radius $$0.4\ \mathrm{m}$$ and mass $$2\ \mathrm{kg}. $$What is the angular acceleration $$\alpha of $$the disk? (Moment of inertia for a solid disk: $$I = \frac{1}{2}MR^2$)

Accepted Answer

$$50\ \mathrm{$$rad/s^2$$}$$

Question 10

A child pushes a merry-go-round with a constant tangential force, causing it to rotate faster. Which of the following statements is true regarding the work done by the child?

Accepted Answer

The work done is equal to the change in rotational kinetic energy only.

Question 11

A gyroscope precesses because:

Accepted Answer

A torque acts perpendicular to its angular momentum, causing the direction of angular momentum to change.

Question 12

A uniform solid sphere and a uniform solid cylinder, both of mass $$M$$ and radius $$R$$, roll without slipping down an incline. Which reaches the bottom first and why?

Accepted Answer

The sphere, because it has a smaller moment of inertia relative to its mass and radius.

Question 13

Which of the following is the correct expression for the total kinetic energy of a rigid body rolling without slipping?

Accepted Answer

$$K = \frac{1}{2}mv^2$ + \frac{1}{2}I\$$omega^2$$

Question 14

A force $$F is $$applied at an angle $$	heta to a $$lever arm of length $$r. $$Which formula gives the magnitude of the torque produced?

Accepted Answer

$$	au = rF\sin	heta$$

Question 15

A rotating flywheel has a constant magnitude of angular momentum, but its direction changes continuously. What is this phenomenon called?

Accepted Answer

Precession

Question 16

If the net torque acting on a system is zero, which of the following quantities must remain constant?

Accepted Answer

Angular momentum

Question 17

A disk rotates with constant angular acceleration $$\alpha. If $$its initial angular velocity is $$\$$omega_0$$ and it rotates through an angle $$	heta$$ in time t$, which equation relates these quantities?

Accepted Answer

$$	heta = \$$omega_0$$ t + \frac{1}{2}\alpha t^2$

Question 18

Which of the following best describes the difference between torque and work?

Accepted Answer

Torque involves force and distance that are perpendicular, while work involves force and distance that are parallel.

Dynamics of Rotational Motion: Torque, Angular Momentum, and Energy

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