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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 $360\mathrm{°}$ 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 $\frac{\text{radians}}{\text{s}}$.
Centripetal acceleration (${a}_{c}$)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 $\frac{\text{m}}{{\text{s}}^{2}}$.
Period ($T$)Time needed for one revolution. Inversely proportional to frequency. SI units of $\text{s}$.
Frequency ($f$) Number of revolutions per second for a rotating object. SI units of $\frac{1}{\text{s}}$ or $\text{Hertz (Hz)}$.

## Equations

EquationSymbol breakdownMeaning in words
$\mathrm{\Delta }\theta =\frac{\mathrm{\Delta }s}{r}$$\mathrm{\Delta }\theta$ is the rotation angle, $\mathrm{\Delta }s$ is the distance traveled around a circle, and $r$ is radiusThe change in angle (in radians) is the ratio of distance travelled around the circle to the circle’s radius.
$\overline{\omega }=\frac{\mathrm{\Delta }\theta }{\mathrm{\Delta }t}$$\overline{\omega }$ is the average angular velocity, $\mathrm{\Delta }\theta$ is rotation angle, and $\mathrm{\Delta }t$ is change in timeAverage angular velocity is proportional to angular displacement and inversely proportional to time.
$v=r\omega$$v$ is linear speed, $r$ is radius, $\omega$ is angular speed.Linear speed is proportional to angular speed times radius $r$. Angular speed is the magnitude of the angular velocity.
$T=\frac{2\pi }{\omega }=\frac{1}{f}$$T$ 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 $v=r\omega$.

### 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).

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

## Want to join the conversation?

• why is an object in uniform circular motion expereincing centripetal acceleration ?
• Think about Newton's first law: An object in motion will stay in motion at a constant speed in a straight line unless acted on upon by an outside force. An object that is moving has inertia that causes it to want to stay in motion in a straight line. But if an object is moving in a circle, the velocity is no longer in a straight line. This means that a force must be acting on the object which means that the object must be accelerating. It is this acceleration that we refer to as centripetal acceleration.
• Is angular velocity only related with circular motion?
• angular velocity must exist wherever angular displacement occurs, irrespective of shape of the path.
• Why is angular velocity sometimes expresed in revolutions per minute, isn't that frecuency?
• well, the angular velocity is expressed by the angular displacement over the change in time, so in your case the revolutions would be the angular displacement converted to revolutions, and the time would be in minutes. Although it isn't in the rad/s form, I suppose it is still the angular displacement per time (in minutes).
• how to calculate angel velocity
(1 vote)
• Angel velocity is more of a theology subject than a physics subject C:
But if you want to find angular velocity, simply divide the angle traveled in radians by the time it took to rotate at that angle.
The general equation is as follows: ω = (θ/t) where omega (ω) is in radians per seconds, theta (θ) is in radians, and t is in seconds.
• what is a matching band
(1 vote)
• A marching band is a group of people playing musical instruments while walking/marching. Typically they wear uniforms and play their music in parades or at events.
• i need 3 examples of circular motion please help me!!
• The orbit of the moon is close. Twirling an object at the end of a string. Race cars driving in a circle. Roller coaster going in a loop.
• if you divide velocity (m/s) by the radius (m) you get the angular velocity which is measured (rad/s), is that right ?

(m/s) / (m) = rad/s ?