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Uniform Circular Motion: Definition & Mathematics

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  • 0:02 What Is Uniform…
  • 1:31 Centripetal vs.…
  • 3:00 Equations
  • 4:00 Example Calculation
  • 5:10 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.

After watching this lesson, you will be able to explain what uniform circular motion is, in terms of both acceleration and forces. You will also be able to use equations for centripetal force and acceleration to solve problems. A short quiz will follow.

What Is Uniform Circular Motion?

Circular motion is a part of life. Planets orbit the sun in circular motion. A car screeching as it goes around a corner is also in circular motion. And if you've ever played lawn tennis, the kind with a ball on a string flying around a pole, then you'll have experienced another example of circular motion. If you're going at a constant speed in your circle, then the motion is said to be uniform.

Uniform circular motion is motion in a circle at a constant speed. This happens because of a centripetal force, a force pointing towards the center of a circle. Mathematically, an object in uniform circular motion has a net force towards the center of the circle, an acceleration vector towards the center of the circle, and a velocity tangent to the circle, as shown in this diagram:

uniform circular motion diagram

An interesting thing about circular motion is that it shows very clearly why it's important to know the difference between scalars and vectors. Speed is a scalar, whereas velocity is a vector - velocity has to include a direction, not just a number. The speed of an object in uniform circular motion is constant because after all that's what makes it uniform. But the velocity is always changing. A satellite or car or bird moving in circular motion is constantly changing direction, so their velocity is constantly changing. This shows why an object can have an acceleration even at a constant speed.

Centripetal vs. Centrifugal Force

An object in circular motion is kept in that circle due to a centripetal force. A centripetal force is a force directed towards the center of a circle. But this seems to go contrary to a lot of people's experiences.

Let's say you're in the passenger seat of a car, when it takes a sharp turn to the left. Where are you pushed? If you have a good memory for this sort of thing, you'll probably answer that you're pushed to the right - or in other words, you're pushed towards the outside of the circle. So surely, the force is away from the center of the circle, not towards it. This is the definition of a centrifugal force, a force pointing away from the center of a circle.

But centrifugal forces don't really exist. When you're sat in a car moving in a straight line, your body wants to keep going in a straight line. Newton's 1st Law, which we talk about in another lesson, says that a body in motion stays in motion, a body at rest stays at rest, unless acted upon by an unbalanced force. So, when the car makes the turn, your body wants to keep going straight. Your body goes straight, but the car turns, causing you to smush against the outside of the curve. But the car is actually keeping you inside the circle, so even though you feel the pressure of the car door, the force your body is experiencing is towards the center of the circle - it's centripetal. If it wasn't, you would just keep going in your nice, neat, straight line.

Equations

There are two main equations you need to know about circular motion. The first helps you calculate the size of that centripetal force. It says that the centripetal force, Fc, measured in newtons, is equal to the mass of the object moving in a circle, m, multiplied by the velocity of the object as it goes around the circle, v, measured in meters per second, squared (it's just the velocity that's squared), divided by the radius of the circle, measured in meters.

centripetal force and acceleration equations

And we also have an equation for centripetal acceleration - the size of the acceleration that's also pointed directly towards the center of the circle, that's measured in meters per second per second, or meters per second squared. The equation's pretty similar, but just without the m. It's just the velocity, v, measured in meters per second, squared, divided by the radius of the circle.

But let's go through an example showing just how to use these equations.

Example Calculation

Let's say you're spinning a ball on a string above your head because that's the kind of thing you do for fun. The velocity of the ball is 4 meters per second, the mass of the ball is 0.5 kilograms, and the radius of the circle above your head is 1.5 meters. You're asked to calculate both the centripetal force and centripetal acceleration.

Well, the first thing we do is write down what we know. We know that the velocity of the ball, v, is 4 meters per second, and the mass of the ball, m, is equal to 0.5 kilograms, and we also know that the radius, r, is 1.5 meters. We have all the numbers we need, so we can start plugging some numbers into the equations.

If we plug these numbers into the force equation, we get 0.5 multiplied by 4 squared divided by 1.5. Type that into a calculator, and we get 5.33 newtons.

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