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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 2π radians in a 360° circle or one revolution. Unitless.
Angular velocity (ω) 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.
Centripetal acceleration (ac)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.
Period (T)Time needed for one revolution. Inversely proportional to frequency. SI units of s.
Frequency (f) Number of revolutions per second for a rotating object. SI units of 1s or Hertz (Hz).


EquationSymbol breakdownMeaning in words
Δθ=ΔsrΔθ is the rotation angle, Δs is the distance traveled around a circle, and r is radiusThe change in angle (in radians) is the ratio of distance travelled around the circle to the circle’s radius.
ω¯=ΔθΔtω¯ is the average angular velocity, Δθ is rotation angle, and Δt is change in timeAverage angular velocity is proportional to angular displacement and inversely proportional to time.
v=rωv is linear speed, r is radius, ω is angular speed.Linear speed is proportional to angular speed times radius r. Angular speed is the magnitude of the angular velocity.
T=2πω=1fT is period, ω is angular speed, and f is frequencyPeriod is inversely proportional to angular speed times a factor of 2π, and inversely proportional to frequency.

How to relate angular speed and linear speed

Angular velocity ω measures the amount of rotation per time. It is a vector and has a direction which corresponds to counterclockwise or clockwise motion (Figure 1).
The same letter ω is often used to the represent the angular speed, which is the magnitude of the angular velocity.
Velocity v measures the amount of displacement per time. It is a vector and has a direction (Figure 1).
The same letter v is often used to represent the speed (sometimes called linear speed in these contexts to differentiate it from angular speed), which is the magnitude of the velocity.
The relationship between the speed v and the angular speed ω is given by the relationship v=rω.
Figure 1. Angular velocity vs. linear velocity

Angular speed does not change with radius

Angular speed ω does not change with radius, but linear speed v 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 who travels a greater distance per time, has a greater linear speed than the person closest to the inside. However, the angular speed of every person in the line is the same because they are moving through the same angle in the same amount of time (Figure 2).
Figure 2. Angular speed remains the same regardless of distance from the center, but the linear speed 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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