Mechanics · Quick Reference

Rotational Motion Cheat Sheet

Mechanics · Lesson 14/19

In one line: rotational motion applies the same core ideas as linear motion -- position, velocity, acceleration -- but to objects spinning around an axis instead of moving in a straight line.

Key Ideas

1Angular Displacement, Velocity, and Acceleration. Angular displacement (theta) measures rotation angle; angular velocity (omega) is the rate of change of theta; angular acceleration (alpha) is the rate of change of omega.

2Rotational Equations of Motion. Mirror the linear equations: omega = omega0 + alpha*t; theta = omega0*t + (1/2)*alpha*t^2 -- same structure as linear motion, with rotational variables substituted in.

3Relating Linear and Angular Quantities. v = omega*r (linear speed at radius r), a_tangential = alpha*r.

4Moment of Inertia. The rotational equivalent of mass -- a measure of an object's resistance to changes in rotational motion, depending on both mass and how that mass is distributed relative to the axis.

5Rotational Kinetic Energy. KE_rotational = (1/2)*I*omega^2, where I is the moment of inertia -- directly analogous to (1/2)*m*v^2 for linear motion.

Worked Examples

A wheel starts at rest and reaches angular velocity 20 rad/s after 4 seconds. Find its angular acceleration.

5 rad/s^2

A wheel of radius 0.5m spins at 10 rad/s. Find the linear speed of a point on its rim.

5 m/s

A wheel with moment of inertia 2 kg*m^2 spins at 6 rad/s. Find its rotational kinetic energy.

36 J

Formulas

omega = omega0 + alpha*t

theta = omega0*t + (1/2)*alpha*t^2

v = omega*r

KE_rotational = (1/2)*I*omega^2

Practice Yourself

A wheel accelerates from rest to 15 rad/s in 3s. Find angular acceleration.

5 rad/s^2

A wheel of radius 0.4m spins at 20 rad/s. Find the linear speed at its rim.

8 m/s

A wheel has I=4 kg*m^2 and omega=3 rad/s. Find rotational KE.

18 J