Q: If a constant external torque $$\tau_{ext} is $$applied to a rigid body over an angular displacement $$\Delta \vartheta$$, what is the change in rotational kinetic energy?

$$\Delta K_{rot} = \left( \sum \tau_{ext} \right) \Delta \vartheta$$

Question 1

Which of the following correctly expresses the moment of inertia $$I$$ for a system of discrete point masses?

Accepted Answer

$$I = \sum_n m_n r_n^2$

Question 2

For an extended object, which integral gives the rotational inertia about a given axis?

Accepted Answer

$$I = \int r^2$ dm$$

Question 3

The parallel axis theorem relates the rotational inertia about any axis to that about a parallel axis through the center of mass. Which equation represents this theorem?

Accepted Answer

$$I = I_{cm} + m d^2$

Question 4

A uniform rod of mass $$m$$ and length $$l$$ rotates about one end. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{3} m l^2$

Question 5

A solid sphere and a hollow sphere of equal mass and radius roll down the same ramp without slipping. Which arrives first and why?

Accepted Answer

The solid sphere arrives first because it has a smaller moment of inertia.

Question 6

The rotational kinetic energy of a rigid body rotating with angular velocity $$\omega$$ and moment of inertia $$I is $$given by:

Accepted Answer

$$K_{rot} = \frac{1}{2} I \$$omega^2$$

Question 7

A rolling object’s total kinetic energy is the sum of translational and rotational kinetic energies. Which formula correctly expresses this?

Accepted Answer

$$K = \frac{1}{2} m v_{cm}^2 + \frac{1}{2} I \$$omega^2$$

Question 8

When a wheel rolls without slipping on a flat surface, what is the velocity of the point of contact with the ground relative to the ground?

Accepted Answer

Zero

Question 9

A disk of mass $$m$$ and radius $$R$$ rolls without slipping. What is the velocity of the topmost point of the disk relative to the ground?

Accepted Answer

$$2 v_{cm}$$

Question 10

If a constant external torque $$	au_{ext} is $$applied to a rigid body over an angular displacement $$\Delta \vartheta$$, what is the change in rotational kinetic energy?

Accepted Answer

$$\Delta K_{rot} = \left( \sum 	au_{ext} ight) \Delta \vartheta$$

Question 11

A ball rolls down an incline without slipping. Which statement best describes the energy transformation?

Accepted Answer

Potential energy is converted into both translational and rotational kinetic energy.

Question 12

For a rolling object, which condition must be satisfied for rolling without slipping?

Accepted Answer

$$v_{cm} = R \omega$$

Question 13

A circular shape is made from three identical balls connected by rods of negligible mass, forming a triangle. If the axes a, b, and c are as shown, which ranking of moments of inertia is correct?

Accepted Answer

$$I_a \u003c I_b \u003c I_c$$

Question 14

Which of the following best explains why friction is necessary for a ball to roll down a ramp without slipping?

Accepted Answer

Friction provides the torque needed to generate rotational kinetic energy.

Question 15

A solid cylinder and a hollow cylinder of equal mass and radius roll down an incline. Which has the greater rotational kinetic energy at the bottom?

Accepted Answer

The hollow cylinder

Question 16

If a wheel of radius $$R is $$rolling without slipping with angular velocity $$\omega$$, what is the velocity of the center of mass?

Accepted Answer

$$v_{cm} = R \omega$$

Question 17

Which of the following is true about the velocity of points on the rim of a rolling wheel (without slipping)?

Accepted Answer

The top point moves at $$2 v_{cm}$$, the bottom point at $$0$$, and the center at $$v_{cm}.$$

Question 18

A uniform rod of mass $$m$$ and length $$l$$ rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{12} m l^2$

Question 19

Which of the following statements about pure rotation is correct?

Accepted Answer

All points on the object move in circles about the axis, and the center of mass does not translate.

Rotational Inertia and Rolling Motion

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