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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

Learn more

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?

  • blobby green style avatar for user aisha
    why is an object in uniform circular motion expereincing centripetal acceleration ?
    (3 votes)
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    • hopper cool style avatar for user obiwan kenobi
      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.
      (9 votes)
  • blobby green style avatar for user abkhanayan.02
    Is angular velocity only related with circular motion?
    (4 votes)
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  • blobby green style avatar for user Milton Jimenez
    Why is angular velocity sometimes expresed in revolutions per minute, isn't that frecuency?
    (4 votes)
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    • mr pants teal style avatar for user Le, Donna
      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).
      (2 votes)
  • blobby green style avatar for user manase0322
    how to calculate angel velocity
    (1 vote)
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    • duskpin ultimate style avatar for user BeeGee
      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.
      (7 votes)
  • blobby green style avatar for user maxou27
    what is a matching band
    (1 vote)
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  • blobby green style avatar for user 😊
    i need 3 examples of circular motion please help me!!
    (0 votes)
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  • blobby green style avatar for user Jonathan  Alejos
    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 ?
    (2 votes)
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  • leaf green style avatar for user Bhavashruthi Mylsamy
    isn't v= cross product of r and w?? then why is v given as speed instead of velocity??
    (2 votes)
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  • blobby green style avatar for user manisharawlani8423
    To move a body in a circle which of th following forces in neend
    (0 votes)
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  • blobby green style avatar for user Винсент
    I feel weird about the sentence "The same letter omega is often used to the represent the angular speed, which is the magnitude of the angular velocity.".
    (1 vote)
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    • blobby green style avatar for user onetwothree
      The same letter is used, but angular velocity is a vector quantity - ie: it has a direction AND a magnitude - and it is denoted by omega with a vector sign on top, while angular speed is a scalar quantity - ie: it only has a magnitude, and it is denoted with no sign on top. Since in the case of angular measures, the only direction is +ve or -ve, in effect angular speed is the absolute value of angular velocity. Hope this helps....
      (1 vote)