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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 2, pi radians in a 360, degree circle or one revolution. Unitless.
Angular velocity (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 start fraction, start text, r, a, d, i, a, n, s, end text, divided by, start text, s, end text, end fraction.
Centripetal acceleration (a, start subscript, c, end subscript)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 start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction.
Period (T)Time needed for one revolution. Inversely proportional to frequency. SI units of start text, s, end text.
Frequency (f) Number of revolutions per second for a rotating object. SI units of start fraction, 1, divided by, start text, s, end text, end fraction or start text, H, e, r, t, z, space, left parenthesis, H, z, right parenthesis, end text.

## Equations

EquationSymbol breakdownMeaning in words
delta, theta, equals, start fraction, delta, s, divided by, r, end fractiondelta, theta is the rotation angle, delta, s 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.
omega, with, \bar, on top, equals, start fraction, delta, theta, divided by, delta, t, end fractionomega, with, \bar, on top is the average angular velocity, delta, theta is rotation angle, and delta, t is change in timeAverage angular velocity is proportional to angular displacement and inversely proportional to time.
v, equals, r, omegav is linear speed, r is radius, omega is angular speed.Linear speed is proportional to angular speed times radius r. Angular speed is the magnitude of the angular velocity.
T, equals, start fraction, 2, pi, divided by, omega, end fraction, equals, start fraction, 1, divided by, f, end fractionT is period, omega is angular speed, and f is frequencyPeriod is inversely proportional to angular speed times a factor of 2, pi, and inversely proportional to frequency.

## How to relate angular speed and linear speed

Angular velocity omega 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 omega 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 omega is given by the relationship v, equals, r, omega.
Figure 1. Angular velocity vs. linear velocity

### Angular speed does not change with radius

Angular speed omega 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