Mechanics · Quick Reference

Circular Motion Cheat Sheet

Mechanics · Lesson 8/19

In one line: an object moving in a circle is constantly changing direction, even at constant speed — this requires a special kind of acceleration and force pointed toward the circle's center.

Key Ideas

1Uniform Circular Motion. Motion in a circle at constant speed; even though speed doesn't change, velocity does (since direction is always changing).

2Centripetal Acceleration. The acceleration directed toward the center of the circle, required to keep an object moving in a circular path: a_c = v^2/r.

3Centripetal Force. The net force directed toward the center that causes centripetal acceleration: F_c = m*v^2/r. This isn't a new type of force -- it's whatever force (tension, gravity, friction) happens to point inward.

4Period and Frequency. Period (T) is the time for one full revolution; frequency (f) is the number of revolutions per second; they're related by f = 1/T.

5Angular Velocity. The rate of change of angle over time, omega = 2*pi/T (in radians per second), related to linear speed by v = omega*r.

Worked Examples

A car moves at 20 m/s around a curve with radius 40m. Find its centripetal acceleration.

10 m/s^2

A 2 kg ball moves in a circle of radius 0.5m at 4 m/s. Find the centripetal force required.

64 N

An object completes one revolution every 4 seconds. Find its angular velocity.

omega = pi/2 radians/second (approximately 1.57 rad/s)

Formulas

a_c = v^2/r

F_c = m*v^2/r

omega = 2*pi/T

v = omega*r

Practice Yourself

A car moves at 10 m/s around a curve of radius 25m. Find centripetal acceleration.

4 m/s^2

A 1 kg ball moves in a circle of radius 2m at 6 m/s. Find centripetal force.

18 N

An object has period 2s. Find its frequency.

0.5 Hz