Uniform Circular Motion Cheat Sheet

Key Facts

• Tangential speed in uniform circular motion is $v = \frac{2\pi r}{T}$, where $r$ is radius and $T$ is period.
• Frequency and period are related by $f = \frac{1}{T}$ and $T = \frac{1}{f}$.
• Tangential speed can also be written as $v = 2\pi r f$ when frequency $f$ is known.
• Centripetal acceleration is $a_c = \frac{v^2}{r}$ and always points toward the center of the circle.
• Centripetal acceleration can also be written as $a_c = \frac{4\pi^2 r}{T^2}$ or $a_c = 4\pi^2 r f^2$.
• The net inward force required for circular motion is $F_c = ma_c = \frac{mv^2}{r}$.
• Velocity is always tangent to the circular path, while acceleration and net force are always radial and inward.
• Uniform circular motion has constant speed but changing velocity because velocity includes direction.

Vocabulary

Uniform circular motion

Motion in a circular path at constant speed, with velocity continuously changing direction.

Tangential speed

The linear speed of an object along the edge of a circle, given by $v = \frac{2\pi r}{T}$.

Centripetal acceleration

The inward acceleration needed to keep an object moving in a circle, given by $a_c = \frac{v^2}{r}$.

Centripetal force

The net inward force that causes centripetal acceleration, given by $F_c = \frac{mv^2}{r}$.

Period

The time for one complete revolution, represented by $T$ and measured in seconds.

Frequency

The number of revolutions per second, represented by $f$ and measured in hertz.