Rotational Motion and Torque — Practice Quiz

A Physics cheat sheet for Rotational Motion and Torque — every key formula with its symbols defined — plus a medium-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 car travels along a circular track with a radius of $50 \text{ m}$. If the car covers an arc length of $150 \text{ m}$, what is its angular displacement?

    • $3 \text{ rad}$
    • $0.33 \text{ rad}$
    • $7500 \text{ rad}$
    • $150 \text{ rad}$

    Answer: $3 \text{ rad}$

  2. A rotating disk changes its angular position from $10 \text{ rad}$ to $70 \text{ rad}$ in $5 \text{ s}$. What is its average angular velocity?

    • $12 \text{ rad/s}$
    • $14 \text{ rad/s}$
    • $16 \text{ rad/s}$
    • $60 \text{ rad/s}$

    Answer: $12 \text{ rad/s}$

  3. A flywheel starts from rest and accelerates uniformly at $2 \text{ rad/s}^2$ for $10 \text{ s}$. What is its final angular velocity?

    • $10 \text{ rad/s}$
    • $20 \text{ rad/s}$
    • $5 \text{ rad/s}$
    • $0 \text{ rad/s}$

    Answer: $20 \text{ rad/s}$

  4. A wheel initially rotating at $5 \text{ rad/s}$ undergoes an angular acceleration of $3 \text{ rad/s}^2$. What is its angular displacement after $4 \text{ s}$?

    • $20 \text{ rad}$
    • $24 \text{ rad}$
    • $44 \text{ rad}$
    • $52 \text{ rad}$

    Answer: $44 \text{ rad}$

  5. A point on the rim of a spinning wheel is $0.5 \text{ m}$ from the center. If the wheel has an angular velocity of $10 \text{ rad/s}$, what is the tangential velocity of this point?

    • $5 \text{ m/s}$
    • $10 \text{ m/s}$
    • $20 \text{ m/s}$
    • $0.5 \text{ m/s}$

    Answer: $5 \text{ m/s}$

  6. A force of $20 \text{ N}$ is applied to a wrench at a distance of $0.3 \text{ m}$ from the pivot. If the force is applied perpendicular to the wrench, what is the magnitude of the torque produced?

    • $6 \text{ N} \cdot \text{m}$
    • $60 \text{ N} \cdot \text{m}$
    • $0.6 \text{ N} \cdot \text{m}$
    • $20.3 \text{ N} \cdot \text{m}$

    Answer: $6 \text{ N} \cdot \text{m}$

  7. A grinding wheel has a moment of inertia of $0.5 \text{ kg} \cdot \text{m}^2$. If a net torque of $10 \text{ N} \cdot \text{m}$ is applied to it, what is its angular acceleration?

    • $5 \text{ rad/s}^2$
    • $10 \text{ rad/s}^2$
    • $20 \text{ rad/s}^2$
    • $0.05 \text{ rad/s}^2$

    Answer: $20 \text{ rad/s}^2$

  8. A solid cylinder with a moment of inertia of $2 \text{ kg} \cdot \text{m}^2$ rotates at an angular velocity of $4 \text{ rad/s}$. What is its rotational kinetic energy?

    • $8 \text{ J}$
    • $16 \text{ J}$
    • $32 \text{ J}$
    • $4 \text{ J}$

    Answer: $16 \text{ J}$

  9. A figure skater has a moment of inertia of $3 \text{ kg} \cdot \text{m}^2$ when spinning at $5 \text{ rad/s}$. What is her angular momentum?

    • $1.67 \text{ kg} \cdot \text{m}^2/\text{s}$
    • $15 \text{ kg} \cdot \text{m}^2/\text{s}$
    • $0.6 \text{ kg} \cdot \text{m}^2/\text{s}$
    • $8 \text{ kg} \cdot \text{m}^2/\text{s}$

    Answer: $15 \text{ kg} \cdot \text{m}^2/\text{s}$

  10. A spinning ice skater pulls her arms in, reducing her moment of inertia from $4 \text{ kg} \cdot \text{m}^2$ to $1 \text{ kg} \cdot \text{m}^2$. If her initial angular velocity was $2 \text{ rad/s}$, what is her final angular velocity?

    • $0.5 \text{ rad/s}$
    • $2 \text{ rad/s}$
    • $4 \text{ rad/s}$
    • $8 \text{ rad/s}$

    Answer: $8 \text{ rad/s}$

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