Rotational Motion and Torque — Hard Practice Quiz

A Physics cheat sheet for Rotational Motion and Torque — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Angular Displacement: Angle \(\theta\) (in radians) is arc length \(s\) divided by radius \(r\).

$$\theta = \frac{s}{r}$$

Average Angular Velocity: Change in angular displacement \(\Delta\theta\) over time \(\Delta t\).

$$\bar{\omega} = \frac{\Delta\theta}{\Delta t}$$

Average Angular Acceleration: Change in angular velocity \(\Delta\omega\) over time \(\Delta t\).

$$\bar{\alpha} = \frac{\Delta\omega}{\Delta t}$$

Rotational Kinematics (Velocity-Time): Final angular velocity \(\omega\) given initial \(\omega_0\), acceleration \(\alpha\), and time \(t\).

$$\omega = \omega_0 + \alpha t$$

Rotational Kinematics (Displacement-Time): Angular displacement \(\theta\) given initial velocity, acceleration, and time.

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

Rotational Kinematics (Velocity-Displacement): Relates velocities, acceleration, and displacement without time.

$$\omega^2 = \omega_0^2 + 2\alpha\theta$$

Tangential Velocity: Linear speed \(v_T\) of a point at radius \(r\) with angular velocity \(\omega\).

$$v_T = r\omega$$

Tangential Acceleration: Linear acceleration \(a_T\) of a point at radius \(r\) with angular acceleration \(\alpha\).

$$a_T = r\alpha$$

Centripetal Acceleration: Radial acceleration component toward the center.

$$a_c = r\omega^2$$

Rolling Motion (No Slipping): Linear velocity and acceleration of the center of mass related to angular quantities.

$$v = r\omega, a = r\alpha$$

Torque: The magnitude of torque \(\tau\) is the distance \(r\) from the pivot to the force \(F\), multiplied by the perpendicular component of the force.

$$\tau = r F \sin(\theta)$$

Rotational Analog of Newton's Second Law: Net torque \(\tau_{net}\) equals moment of inertia \(I\) multiplied by angular acceleration \(\alpha\).

$$\tau_{net} = I\alpha$$

Moment of Inertia (Point Masses): The moment of inertia \(I\) is the sum of each mass \(m\) times its distance \(r\) from the axis squared.

$$I = \sum mr^2$$

Angular Velocity (Constant Angular Acceleration): Final angular velocity \(\omega_f\) equals initial \(\omega_i\) plus angular acceleration \(\alpha\) times time \(t\).

$$\omega_f = \omega_i + \alpha t$$

Rotational Kinetic Energy: The energy of an object due to its rotation.

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

Angular Momentum: Angular momentum \(L\) is the moment of inertia \(I\) multiplied by the angular velocity \(\omega\).

$$L = I\omega$$

Net Torque and Angular Momentum: Net external torque equals the rate of change of angular momentum.

$$\sum \tau = \frac{\Delta L}{\Delta t}$$

Equilibrium Conditions: For a rigid body to be in equilibrium, the net force and net torque must both be zero.

$$\sum F_x = 0, \sum F_y = 0, \sum \tau = 0$$

Center of Gravity: The point where the total weight of the body can be considered to act.

$$x_{cg} = \frac{W_1 x_1 + W_2 x_2 + \dots}{W_1 + W_2 + \dots}$$

Rotational Work: Work done by a constant torque \(\tau\) rotating an object through angle \(\theta\).

$$W_R = \tau \theta$$

Total Mechanical Energy (Rolling): Sum of translational KE, rotational KE, and gravitational PE.

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

Conservation of Angular Momentum: If net external torque is zero, total angular momentum is conserved.

$$L_f = L_0$$

Practice quiz

  1. A wheel starts from rest and accelerates with constant angular acceleration $\alpha$. At time $t$, a point on its rim has tangential acceleration $a_T$ and centripetal acceleration $a_c$. If the angular acceleration is doubled to $2\alpha$, what is the ratio of the new centripetal acceleration to the original centripetal acceleration at the same time $t$?

    • A) $2$
    • B) $4$
    • C) $1/2$
    • D) $1/4$

    Answer: B) $4$

  2. A uniform solid disk of mass $M$ and radius $R$ is rotating with angular velocity $\omega_0$ about an axis through its center. A second identical disk, initially at rest, is dropped onto the first disk. The two disks eventually rotate together with a common final angular velocity $\omega_f$. What fraction of the initial rotational kinetic energy is lost during this process?

    • A) $1/4$
    • B) $1/2$
    • C) $2/3$
    • D) $3/4$

    Answer: B) $1/2$

  3. A uniform rod of length $L$ and mass $M$ is pivoted at one end and released from rest in a horizontal position. What is the initial angular acceleration of the rod? (Moment of inertia of a rod about one end is $I = \frac{1}{3}ML^2$).

    • A) $\frac{3g}{2L}$
    • B) $\frac{g}{L}$
    • C) $\frac{2g}{3L}$
    • D) $\frac{g}{2L}$

    Answer: A) $\frac{3g}{2L}$

  4. A solid sphere of mass $M$ and radius $R$ rolls without slipping down an incline of height $h$. What is its speed at the bottom of the incline? (Moment of inertia of a solid sphere is $I = \frac{2}{5}MR^2$).

    • A) $\sqrt{\frac{2}{5}gh}$
    • B) $\sqrt{\frac{10}{7}gh}$
    • C) $\sqrt{\frac{7}{10}gh}$
    • D) $\sqrt{2gh}$

    Answer: B) $\sqrt{\frac{10}{7}gh}$

  5. A uniform ladder of length $L$ and mass $M$ rests against a frictionless wall at an angle $\theta$ with the horizontal ground. The coefficient of static friction between the ladder and the ground is $\mu_s$. What is the minimum angle $\theta$ for the ladder to remain in equilibrium?

    • A) $\arctan(\frac{1}{\mu_s})$
    • B) $\arctan(\frac{1}{2\mu_s})$
    • C) $\arctan(2\mu_s)$
    • D) $\arctan(\mu_s)$

    Answer: B) $\arctan(\frac{1}{2\mu_s})$

  6. A figure skater is spinning with angular velocity $\omega_0$ and moment of inertia $I_0$. She pulls her arms in, reducing her moment of inertia to $I_f = I_0/3$. What is the ratio of her final rotational kinetic energy to her initial rotational kinetic energy?

    • A) $1/3$
    • B) $1$
    • C) $3$
    • D) $9$

    Answer: C) $3$

  7. A merry-go-round starts from rest and accelerates with a constant angular acceleration of $\alpha = 0.05 \text{ rad/s}^2$. A child is sitting at a distance $r = 2 \text{ m}$ from the center. How many revolutions has the merry-go-round completed when the child's tangential speed reaches $v_T = 1.5 \text{ m/s}$?

    • A) $0.447$ revolutions
    • B) $0.895$ revolutions
    • C) $1.79$ revolutions
    • D) $2.68$ revolutions

    Answer: B) $0.895$ revolutions

  8. A uniform rod of mass $M$ and length $L$ is free to rotate about a pivot at its center. A force $F$ is applied perpendicularly to one end of the rod. If the rod's moment of inertia about its center is $I = \frac{1}{12}ML^2$, derive an expression for the angular acceleration $\alpha$ in terms of $F$, $M$, and $L$.

    • A) $\frac{6F}{ML}$
    • B) $\frac{12F}{ML}$
    • C) $\frac{2F}{ML}$
    • D) $\frac{F}{ML}$

    Answer: A) $\frac{6F}{ML}$

  9. A uniform plank of length $L$ and mass $M$ is supported at two points. One support is at the left end, and the other is at a distance $x$ from the left end. A person of mass $m$ stands at the right end of the plank. If the plank is just about to tip, what is the distance $x$ of the second support from the left end?

    • A) $\frac{L(M/2 + m)}{M+m}$
    • B) $\frac{L(M+m)}{M/2 + m}$
    • C) $\frac{L(M/2)}{M+m}$
    • D) $\frac{L(m)}{M+m}$

    Answer: A) $\frac{L(M/2 + m)}{M+m}$

  10. A flywheel with a moment of inertia $I = 2.0 \text{ kg} \cdot \text{m}^2$ is initially rotating at $\omega_0 = 10 \text{ rad/s}$. A constant braking torque of $\tau = 5.0 \text{ N} \cdot \text{m}$ is applied. How many revolutions does the flywheel make before coming to rest?

    • A) $1.59$ revolutions
    • B) $3.18$ revolutions
    • C) $6.37$ revolutions
    • D) $10.0$ revolutions

    Answer: B) $3.18$ revolutions

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