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Centripetal Acceleration: Definition, Formula & Example

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  • 0:00 What Is Centripetal…
  • 1:48 Equation
  • 2:21 Example
  • 3:01 Lesson Summary
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Lesson Transcript
Instructor
David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

In this lesson, you will learn the meaning of the term centripetal acceleration, an equation to calculate it, and an example of how to use the equation. A short quiz will follow.

What Is Centripetal Acceleration?

If you whirl a ball on a string over your head, the ball is undergoing centripetal acceleration. If you drive your car around in a circle, your car is undergoing centripetal acceleration. And, a satellite orbiting the Earth also has a centripetal acceleration.

Centripetal acceleration is the idea that any object moving in a circle, in something called circular motion, will have an acceleration vector pointed towards the center of that circle. This is true even if the object is moving around the circle at a constant speed. Centripetal means towards the center.

An acceleration is a change in velocity. So, how can something moving at a constant speed in a circle have an acceleration? Well, speed and velocity are not quite the same thing. Speed is just how fast you're going. It is a scalar because it doesn't have a direction. Velocity, on the other hand, is your speed and direction. It is a vector because it does have a direction. For example, 3 miles per hour is a speed, but 3 miles per hour south is a velocity.

Since an object moving in a circle is constantly changing direction, its velocity is constantly changing. And whenever something's velocity is changing - even if only its direction, not its speed - that object must be accelerating.

A force always causes the centripetal acceleration. For a swingball (or tetherball) game, it is the tension in the string. For a satellite, it is the force of gravity. For a car moving around a corner, it is the frictional force between the car and the road. If you remove that force, you remove the centripetal acceleration, and if you remove the centripetal acceleration, the object will continue in a straight line tangent to the circle.

Equation

Acceleration is measured in meters per second per second. This is because it is the number of meters per second by which your velocity changes... each second. For an object moving in circular motion, we can calculate it using the following equation:

In this equation:

a is the centripetal acceleration, measured in meters per second per second (m/s/s)
v is the numerical velocity of the object, measured in meters per second (m/s)
r is the radius of the circle, measured in meters (m)

Example

Let's say you're driving down the street at 50 meters per second, when suddenly you take a corner sharply without slowing down at all. Assuming you manage to stay on the road without hurting yourself, and if the radius of the corner was 5 meters, what was your centripetal acceleration?

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Additional Activities

Calculating Centripetal Acceleration

In the following examples, students will get more experience calculating the centripetal acceleration of an object moving around a circle, given the object's numerical velocity measured in meters per second and the radius of the circle measured in meters. Students will also use the centripetal acceleration equation to find the radius of the circle given the numerical velocity and centripetal acceleration of the object. After completing the beginning examples, students will be ready to try the challenge examples, one of which is theoretical and the other which is more involved.

Beginning Examples

  • Find the centripetal acceleration of an object which is moving around a circle with radius 7 meters at a velocity of 20 meters per second.
  • If the centripetal acceleration of an object is 37 meters per second per second when the object has a velocity of 10 meters per second, what is the radius of the circular path the object is following?

Solutions

  • Using the equation for centripetal acceleration, a=(v^2)/r, we have a=(20^2)/7 = 400/7 so the centripetal acceleration is approximately 57.143 meters per second per second.
  • Using the equation a=(v^2)/r we have 37 = (10^2)/r so r=100/37. The radius of the circle is about 2.703 meters.

Challenge Examples

  • Will the centripetal acceleration for an object with velocity v increase or decrease as the radius of the circle gets larger?
  • What radius will result in the same centripetal acceleration for an object with velocity 50 m/s as for an object with velocity 100 m/s moving around a circle with radius 10 m?

Solutions

  • The equation for centripetal acceleration is a=(v^2)/r. Since we divide by the radius, a larger radius will result in a smaller centripetal acceleration. The centripetal acceleration for a particular object with velocity v will decrease as the radius increases.
  • First find the centripetal acceleration for the faster object. We have a=(100^2)/10 = 1000 m/s/s. We want to find what radius is needed for the slower object in order for the centripetal acceleration to still be 1000 m/s/s. We have 1000=(50^2)/r and so r=2500/1000 = 2.5 meters. The centripetal acceleration for an object moving at 100 m/s around a circle of radius 10 m is equal to the centripetal acceleration for an object moving at 50 m/s around a circle of radius 2.5 meters.

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