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Torque and Rotational Motion Cheat Sheet

The core ideas of Torque and Rotational Motion distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Torque

The rotational equivalent of force. Torque equals the lever arm times the component of force perpendicular to the lever arm: tau = rF sin(theta). It causes angular acceleration around an axis of rotation. Measured in newton-meters (N*m).

Moment of Inertia

The rotational equivalent of mass — a measure of an object's resistance to angular acceleration. It depends on both the mass and how that mass is distributed relative to the axis of rotation: I = sum(m_i * r_i^2).

Angular Momentum

The rotational equivalent of linear momentum, defined as L = I * omega for a rigid body rotating about a fixed axis. Angular momentum is conserved when no net external torque acts on the system.

Newton's Second Law for Rotation

The net torque on an object equals its moment of inertia times its angular acceleration: tau_net = I * alpha. This is the rotational analog of F_net = ma.

Rotational Kinetic Energy

The kinetic energy of a rotating object: KE_rot = (1/2)I * omega^2. For rolling objects, total KE = (1/2)mv^2 + (1/2)I * omega^2 (translational + rotational).

Static Equilibrium

A condition where both the net force and the net torque on an object are zero, so the object is neither accelerating linearly nor angularly. Essential for analyzing beams, ladders, and bridges.

Rolling Without Slipping

A condition where the contact point between a rolling object and the surface has zero velocity. The translational speed v and angular speed omega are related by v = R * omega.

Angular Velocity and Angular Acceleration

Angular velocity (omega) is the rate of change of angular position, measured in rad/s. Angular acceleration (alpha) is the rate of change of angular velocity, measured in rad/s^2.

Parallel Axis Theorem

The moment of inertia about any axis parallel to an axis through the center of mass equals the center-of-mass moment of inertia plus M*d^2, where d is the distance between the axes: I = I_cm + Md^2.

Key Terms at a Glance

Torque:The rotational equivalent of force. tau = r x F. Causes angular acceleration. Unit: N*m.
Moment of Inertia:Rotational equivalent of mass. I = sum(m*r^2). Resistance to angular acceleration. Depends on mass distribution.
Angular Momentum:L = I*omega for a rigid body. Conserved when net external torque is zero. Rotational analog of linear momentum.
Angular Velocity:Rate of change of angular position. omega = d(theta)/dt. Unit: rad/s.
Angular Acceleration:Rate of change of angular velocity. alpha = d(omega)/dt. Unit: rad/s^2.
Rotational Kinetic Energy:KE_rot = (1/2)*I*omega^2. Energy due to rotation about an axis.
Static Equilibrium:Net force = 0 and net torque = 0. Object has no translational or rotational acceleration.
Rolling Without Slipping:v = R*omega. The contact point is instantaneously at rest. No kinetic friction; static friction enables rolling.
Lever Arm:The perpendicular distance from the axis of rotation to the line of action of the force. Determines torque magnitude.
Parallel Axis Theorem:I = I_cm + Md^2. Relates moment of inertia about any axis to that about a parallel axis through the center of mass.
Right-Hand Rule:A method for determining the direction of angular velocity, torque, or angular momentum vectors in rotational motion.
Rotational Inertia:Another name for moment of inertia. An object's tendency to resist changes in its rotational state of motion.

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Torque and Rotational Motion Cheat Sheet - Quick Reference Guide | PiqCue