Rotation and Rotational Dynamics Practice Test

1.

A uniform rod of length L and mass M pivots about one end. If it is released from rest at an angle θ above the horizontal, derive an expression for its angular velocity as it passes through the vertical position. (Neglect air resistance.)

a) \(\sqrt{\frac{3g(1-\cos\theta)}{L}}\) b) \(\sqrt{\frac{3gL\cos\theta}{2}}\) c) \(\sqrt{2gL(1-\cos\theta)}\) d) \(\sqrt{gL(1-\cos\theta)}\)

2.

A rod of length 1.2 m and mass 3 kg rotates about one end. If a force of 6 N is applied perpendicular to the free end, what is the angular acceleration?

a) 3.47 rad/s² b) 5.00 rad/s² c) 2.08 rad/s² d) 1.67 rad/s²

3.

A wheel (I = 0.5 kg·m²) is spinning at 12 rad/s. A brake applies a constant torque of 3 N·m. How long does it take to stop?

a) 2 s b) 3 s c) 4 s d) 6 s

4.

A solid sphere (mass 0.8 kg, radius 0.2 m) rolls without slipping down a 1.5 m high incline. What is its speed at the bottom? (You must use energy conservation and the moment of inertia for a sphere.)

a) 4.4 m/s b) 5.4 m/s c) 3.8 m/s d) 2.7 m/s

5.

A force F is applied tangentially to the rim of a stationary disk (radius R, mass M) for time t. What is the final angular velocity? (You must derive your answer using Newton's second law for rotation and the moment of inertia for a disk.)

a) \(\omega = \frac{2Ft}{MR}\) b) \(\omega = \frac{Ft}{MR^2}\) c) \(\omega = \frac{Ft}{2MR^2}\) d) \(\omega = \frac{Ft}{MR}\)

6.

A yo-yo (mass 0.2 kg, radius 0.04 m) unwinds from rest. If it descends 1.2 m, what is its angular speed at the bottom? (Assume I = 0.5MR², g = 10 m/s². You must use energy conservation and relate linear and angular variables.)

a) 24.5 rad/s b) 27.4 rad/s c) 34.6 rad/s d) 39.2 rad/s

7.

A disk (I = 0.8 kg·m²) is spinning at 6 rad/s. A constant torque of 1.6 N·m is applied in the direction of rotation for 4 seconds. What is its final angular velocity? (Show your reasoning using rotational kinematics.)

a) 14 rad/s b) 10 rad/s c) 8 rad/s d) 12 rad/s

8.

A uniform rod (length 2 m, mass 4 kg) is pivoted at one end and held horizontally. What is the initial torque due to gravity about the pivot? (You must use the center of mass location in your calculation.)

a) 39.2 N·m b) 19.6 N·m c) 9.8 N·m d) 78.4 N·m

9.

A solid cylinder (mass 1.5 kg, radius 0.3 m) rolls without slipping at 4 m/s. What is its total kinetic energy? (You must combine translational and rotational kinetic energy.)

a) 18 J b) 12 J c) 9 J d) 6 J

10.

A flywheel (I = 2 kg·m²) is rotating at 30 rad/s. How much work is required to bring it to rest? (You must use the work-energy theorem for rotation.)

a) 900 J b) 600 J c) 1200 J d) 1800 J

11.

A disk (I) is spinning at \(\omega_0\). A constant torque \(\tau\) is applied in the opposite direction for time t, bringing it to rest. What is the total angle (in radians) through which the disk turns before stopping?

a) \(\frac{I\omega_0^2}{2\tau}\) b) \(\frac{I\omega_0}{\tau}\) c) \(\frac{I\omega_0^2}{\tau}\) d) \(\frac{I\omega_0}{2\tau}\)

12.

A uniform rod (length L, mass m) is pivoted at its center. Two equal and opposite forces F are applied at the ends, perpendicular to the rod. What is the angular acceleration?

a) \(\frac{12F}{mL}\) b) \(\frac{6F}{mL}\) c) \(\frac{3F}{mL}\) d) \(\frac{F}{mL}\)

13.

A solid sphere (I = 2/5 mR^2) is rolling without slipping. If its center of mass has speed v, what is its total kinetic energy?

a) \(\frac{7}{10}mv^2\) b) \(\frac{1}{2}mv^2\) c) \(mv^2\) d) \(\frac{5}{7}mv^2\)

14.

A particle of mass m moves in a circle of radius R under a central force F = -kr. What is the angular frequency of small oscillations about the equilibrium radius?

a) \(\sqrt{\frac{k}{m}}\) b) \(\frac{k}{m}\) c) \(\sqrt{\frac{m}{k}}\) d) \(\frac{m}{k}\)

15.

A disk (I) is spinning at \(\omega_0\). A constant frictional torque \(\tau_f\) acts for time t, bringing it to rest. What is the work done by friction?

a) \(-\frac{1}{2}I\omega_0^2\) b) \(-I\omega_0^2\) c) \(-\tau_f t\) d) \(-I\omega_0\)

16.

A uniform disk (I = ½MR²) of mass 3 kg and radius 0.4 m is spun up from rest by a constant torque of 2.4 N·m. How long does it take to reach 30 rad/s?

a) 4 s b) 5 s c) 6 s d) 8 s

17.

A solid cylinder (mass 2 kg, radius 0.3 m) is spinning at 10 rad/s. A time-dependent torque \(\tau(t) = 3t\) N·m is applied for 2 seconds. What is the cylinder's angular velocity at t = 2 s? (Assume I = 0.5MR² and show your calculus-based reasoning.)

a) 34 rad/s b) 22 rad/s c) 28 rad/s d) 40 rad/s

18.

A solid sphere and a hollow sphere (same mass and radius) roll down a ramp. Which reaches the bottom first?

a) The sphere b) The cylinder c) They arrive together d) The one with greater mass

19.

A rod of length L and mass M is pivoted about one end. A force F is applied perpendicular to the free end. Derive the angular acceleration in terms of F, M, and L.

a) \(\frac{3F}{ML}\) b) \(\frac{F}{ML}\) c) \(\frac{6F}{ML}\) d) \(\frac{F}{2ML}\)

20.

A particle of mass m moves in a circle of radius R under a central force F = -kr. If the radius is slowly increased, what happens to the angular momentum and kinetic energy? (Assume no external torque.)

a) Angular momentum constant, kinetic energy decreases b) Both increase c) Angular momentum increases, kinetic energy constant d) Both decrease

21.

A wheel (I = 1.2 kg·m²) is initially at rest. A torque \(\tau(t) = 6e^{-t}\) N·m is applied for 4 seconds. What is the angular velocity at t = 4 s? (Use calculus to solve.)

a) 3.8 rad/s b) 4.2 rad/s c) 5.0 rad/s d) 6.0 rad/s

22.

A flywheel (I = 0.6 kg·m²) is spinning at 10 rad/s. A constant torque of -1.2 N·m is applied. How many radians does it rotate before stopping?

a) 25 b) 50 c) 30 d) 60

23.

A solid sphere (I = 2/5 mR²) and a hollow sphere (I = 2/3 mR²) of equal mass and radius roll down the same incline. Which arrives first and why?

a) Solid sphere, because it has a smaller moment of inertia b) Hollow sphere, because it has a larger moment of inertia c) Both arrive at the same time d) It depends on the mass

24.

A disk (I = ½MR²) is rotating at ω₀. A constant power P is applied to increase its speed. What is the time required to double its angular speed?

a) \(\frac{3I\omega_0^2}{2P}\) b) \(\frac{I\omega_0^2}{P}\) c) \(\frac{I\omega_0^2}{2P}\) d) \(\frac{2I\omega_0^2}{P}\)

25.

A uniform rod (length L, mass m) is pivoted at one end and released from rest horizontally. What is the linear speed of the free end as it passes through the vertical?

a) \(\sqrt{3gL}\) b) \(\sqrt{2gL}\) c) \(\sqrt{gL}\) d) \(\sqrt{6gL}\)

26.

A disk (I = ½MR²) is rotating at ω₀. A constant torque τ is applied. How much work is done by the torque to triple the angular speed?

a) \(4I\omega_0^2\) b) \(3I\omega_0^2\) c) \(8I\omega_0^2\) d) \(2I\omega_0^2\)

27.

A particle of mass m moves in a circle of radius R with a speed v. If a constant tangential force F is applied, what is the angular acceleration?

a) \(\frac{F}{mR}\) b) \(\frac{F}{mR^2}\) c) \(\frac{F}{vR}\) d) \(\frac{F}{v^2R}\)

28.

A uniform disk and a ring (same mass and radius) are released from rest at the top of an incline. Which reaches the bottom first?

a) The disk b) The ring c) They arrive together d) The one with greater mass

29.

A disk (I = ½MR²) is spinning at ω₀. A constant torque τ is applied in the opposite direction. How long does it take to bring the disk to rest?

a) \(\frac{I\omega_0}{\tau}\) b) \(\frac{2I\omega_0}{\tau}\) c) \(\frac{I\omega_0^2}{\tau}\) d) \(\frac{I\omega_0^2}{2\tau}\)

30.

A system of particles is subject to no external torque. If the moment of inertia decreases by a factor of 4, what happens to the angular speed?

a) It increases by a factor of 4 b) It decreases by a factor of 2 c) It increases by a factor of 2 d) It decreases by a factor of 4

Rotation and Rotational Dynamics Practice Test

30 Questions – Mixed Conceptual & Calculations