Uniform circular motion and centripetal acceleration review

Review the key concepts, equations, and skills for uniform circular motion, including centripetal acceleration and the difference between linear and angular velocity.

Key terms

Term (symbol)Meaning
Uniform circular motionMotion in a circle at a constant speed
RadianRatio of an arc’s length to its radius. There are 2π2\pi radians in a 360°360 \degree circle or one revolution. Unitless.
Angular velocity (ω\omega) Measure of how an angle changes over time. The rotational analogue of linear velocity. Vector quantity with counterclockwise defined as the positive direction. SI units of radianss\dfrac{\text{radians}}{\text {s}}.
Centripetal acceleration (aca_c)Acceleration pointed towards the center of a curved path and perpendicular to the object’s velocity. Causes an object to change its direction and not its speed along a circular pathway. Also called radial acceleration. SI units are ms2\dfrac{\text m}{\text s^2}.
Period (TT)Time needed for one revolution. Inversely proportional to frequency. SI units of s\text{s}.
Frequency (ff) Number of revolutions per second for a rotating object. SI units of 1s\dfrac{1}{\text{s}} or Hertz (Hz)\text{Hertz (Hz)}.

Equations

EquationSymbol breakdownMeaning in words
Δθ=Δsr\Delta \theta = \dfrac{\Delta s}{r}Δθ\Delta \theta is the rotation angle, Δs\Delta s is the distance traveled around a circle, and rr is radiusThe change in angle (in radians) is the ratio of distance travelled around the circle to the circle’s radius.
ωˉ=ΔθΔt\bar \omega = \dfrac{\Delta\theta}{\Delta t}ωˉ\bar \omega is the average angular velocity, Δθ\Delta\theta is rotation angle, and Δt\Delta t is change in timeAverage angular velocity is proportional to angular displacement and inversely proportional to time.
v=rωv = r \omega vv is linear velocity, rr is radius, ω\omega is angular velocityLinear velocity is proportional to angular velocity times radius rr.
T=2πω=1fT = \dfrac{2\pi}{\omega} = \dfrac{1}{f}TT is period, ω\omega is angular velocity, and ff is frequencyPeriod is inversely proportional to angular velocity times a factor of 2π2\pi, and inversely proportional to frequency.

How to relate linear and angular velocity

Angular velocity ω\omega is angular displacement divided by time, while (linear) velocity vv is linear displacement divided by time (Figure 1). Linear velocity is also sometimes called tangential velocity.
Tangential velocity is the straight line velocity of a point on the edge of a rotating object. It is called tangential velocity because the direction of motion is tangent to the circular path. It is the path we could imagine an object taking if we had it on a string and rotating it, but then we let it go.
Figure 1. Angular velocity vs. linear velocity

Angular velocity does not change with radius

Angular velocity does not change with radius, but linear velocity does. For example, in a marching band line going around a corner, the person on the outside has to take the largest steps to keep in line with everyone else. Therefore, the outside person has a much larger linear velocity than the person closest to the inside. However, the angular velocity of every person in the line is the same because they are moving the same angle in the same amount of time (Figure 2).
Figure 2. Angular velocity remains the same regardless of distance from the center, but the linear velocity increases proportionally with radius. Image adapted from Wikimedia Commons. Original image from Wikimedia Commons, CC BY-SA 4.0

Learn more

To check your understanding and work toward mastering these concepts, check out our exercise on calculating angular velocity, period, and frequency.
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