Rotational Motion & Torque Cheat Sheet

A printable reference covering angular kinematics, torque, moment of inertia, rotational dynamics, angular momentum, and rolling motion for grades 11-12.

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Rotational motion describes how objects spin, rotate, roll, and turn about an axis. This cheat sheet helps students connect linear motion ideas to angular quantities such as angular velocity, angular acceleration, torque, and moment of inertia. It is useful for solving problems involving wheels, pulleys, disks, rods, doors, gears, and rotating systems in equilibrium or acceleration. The most important idea is that rotation depends not only on force, but also on where and how the force is applied. Torque follows $\tau = rF\sin\theta$ , rotational dynamics follows $\sum \tau = I\alpha$ , and angular momentum follows $L = I\omega$ . Rolling motion combines translation and rotation using relationships such as $v = r\omega$ and $a = r\alpha$ .

Key Facts

Angular displacement, angular velocity, and angular acceleration are related by $\omega = \frac{\Delta \theta}{\Delta t}$ and $\alpha = \frac{\Delta \omega}{\Delta t}$ .
For constant angular acceleration, the rotational kinematics equations include $\omega_f = \omega_i + \alpha t$ and $\theta = \omega_i t + \frac{1}{2}\alpha t^2$ .
Tangential speed and angular speed are related by $v = r\omega$ , where $r$ is the distance from the rotation axis.
Tangential acceleration and angular acceleration are related by $a_t = r\alpha$ , while centripetal acceleration is $a_c = r\omega^2 = \frac{v^2}{r}$ .
Torque is given by $\tau = rF\sin\theta$ , where $\theta$ is the angle between the lever arm and the applied force.
Newton's second law for rotation is $\sum \tau = I\alpha$ , where $I$ is the moment of inertia and $\alpha$ is angular acceleration.
Rotational kinetic energy is $K_{rot} = \frac{1}{2}I\omega^2$ , and total kinetic energy for rolling without slipping is $K = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$ .
Angular momentum is $L = I\omega$ for a rigid body, and it is conserved when the net external torque is zero, so $L_i = L_f$ .

Vocabulary

Angular Displacement: Angular displacement is the change in rotational position, usually measured in radians as $\Delta \theta$ .
Angular Velocity: Angular velocity is the rate of change of angular displacement, given by $\omega = \frac{\Delta \theta}{\Delta t}$ .
Torque: Torque is the rotational effect of a force and is calculated with $\tau = rF\sin\theta$ .
Moment of Inertia: Moment of inertia is a measure of an object's resistance to angular acceleration, written as $I = \sum mr^2$ for point masses.
Angular Momentum: Angular momentum is rotational momentum, given by $L = I\omega$ for a rigid rotating body.
Rolling Without Slipping: Rolling without slipping occurs when the contact point is instantaneously at rest and the motion satisfies $v = r\omega$ .

Common Mistakes to Avoid

Using force instead of torque: This is wrong because rotation depends on both the force and the lever arm, so the correct quantity is $\tau = rF\sin\theta$ .
Forgetting the angle in torque: This is wrong because only the perpendicular component of force causes rotation, so $\tau$ is not always equal to $rF$ .
Using mass instead of moment of inertia: This is wrong because rotational acceleration depends on mass distribution, so $\sum \tau = I\alpha$ uses $I$ , not just $m$ .
Mixing degrees and radians in angular equations: This is wrong because formulas such as $v = r\omega$ and $a_t = r\alpha$ require angles in radians.
Ignoring rotational kinetic energy in rolling problems: This is wrong because a rolling object has both translational energy and rotational energy, so $K = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$ .

Practice Questions

1 A wheel starts from rest and has angular acceleration $\alpha = 4.0\,\text{rad/s}^2$ for $t = 3.0\,\text{s}$ . Find its final angular velocity $\omega_f$ .
2 A force of $25\,\text{N}$ is applied perpendicular to a wrench at a distance $r = 0.30\,\text{m}$ from the pivot. Find the torque $\tau$ .
3 A solid disk has $I = 0.50\,\text{kg}\cdot\text{m}^2$ and angular speed $\omega = 12\,\text{rad/s}$ . Find its rotational kinetic energy $K_{rot}$ .
4 Two equal forces are applied to a door, one near the hinge and one near the handle. Explain which force creates the larger torque and why.