Any object moving in a circle is accelerating, even if its speed remains the same. This might seem counterintuitive because how can you have acceleration without a change in speed? In fact, because acceleration is the rate of change of velocity, and velocity includes speed and the direction of motion, it’s impossible to have circular motion without acceleration. By Newton’s second law, any acceleration (a) is linked to a force (F) by F = ma, and in the case of circular motion, the force in question is called the centripetal force. Working this out is a simple process, but you might have to think about the situation in different ways depending on the information you have.
TL;DR (Too Long; Didn't Read)
Find the centripetal force using the formula:
F = mv2 / r
Here, F references the force, m is the mass of the object, v is the tangential speed of the object, and r is the radius of the circle it travels in. If you know the source of the centripetal force (gravity, for example), you can find the centripetal force using the equation for that force.
What Is Centripetal Force?
Centripetal force isn’t a force in the same way as gravitational force or frictional force. Centripetal force exists because centripetal acceleration exists, but the physical cause of this force can vary depending on the specific situation.
Consider the Earth’s motion around the sun. Even though the speed of its orbit is constant, it changes direction continuously and therefore has acceleration directed toward the sun. This acceleration must be caused by a force, according to Newton’s first and second laws of motion. In the case of the Earth’s orbit, the force causing the acceleration is gravity.
However, if you swing a ball on a string in a circle at a constant speed, the force causing the acceleration is different. In this case, the force is from the tension in the string. Another example is a car maintaining a constant speed but turning in a circle. In this case, the friction between the car’s wheels and the road is the source of the force.
In other words, centripetal forces exist, but the physical cause of them depends on the situation.
Formula for Centripetal Force and Centripetal Acceleration
Centripetal acceleration is the name for the acceleration directly toward the center of the circle in circular motion. This is defined by:
Where v is the speed of the object in the line tangential to the circle, and r is the radius of the circle it is moving in. Think about what would happen if you were swinging a ball connected to a string in a circle, but the string broke. The ball would fly off in a straight line from its position on the circle at the time the string broke, and this gives you an idea what v means in the above equation.
Because Newton’s second law states that force = mass × acceleration, and we have an equation for acceleration above, the centripetal force must be:
In this equation, m refers to mass.
So, to find the centripetal force, you need to know the mass of the object, the radius of the circle it’s traveling in and its tangential speed. Use the equation above to find the force based on these factors. Square the speed, multiply it by the mass and then divide the result by the radius of the circle.
Angular Velocities: You can also use the angular velocity ω of the object if you know it; it is the rate of change of the object’s angular position with time. This changes the centripetal acceleration equation to:
a = ω2r
The centripetal force equation becomes:
F = mω2r
Finding Centripetal Force With Incomplete Information
If you don’t have all the information you need for the equation above, it might seem like finding the centripetal force is impossible. However, if you think about the situation, you can often work out what the force might be.
For example, if you’re trying to find the centripetal force acting on a planet orbiting a star or a moon orbiting a planet, you know that the centripetal force comes from gravity. This means you can find the centripetal force without the tangential velocity by using the ordinary equation for gravitational force:
Where m1 and m2 are the masses, G is the gravitational constant, and r is the separation between the two masses.
To calculate centripetal force without a radius, you need either more information (the circumference of the circle related to radius by C = 2πr, for example) or the value for the centripetal acceleration. If you know the centripetal acceleration, you can calculate the centripetal force directly using Newton’s second law, F = ma.
About the Author
Lee Johnson is a freelance writer and science enthusiast, with a passion for distilling complex concepts into simple, digestible language. He's written about science for several websites including eHow UK and WiseGeek, mainly covering physics and astronomy. He was also a science blogger for Elements Behavioral Health's blog network for five years. He studied physics at the Open University and graduated in 2018.