# Centripetal Force Equation

Centripetal Force Equation: Do you remember riding on the merry-go-round as a kid? Did you ever stand at the very edge of the merry-go-round and hold on tight to the railing as your friends pushed the wheel faster and faster? Maybe you remember that the faster the wheel turned, the harder it became to hold on. You might not have known it at the time, but you were creating a balance between two forces – one real and one apparent – in order to stay on that circular path.

Merry-go-rounds are a perfect example of how a force is used to keep an object moving in a circular path. Your body wanted to fly off the merry-go-round in a straight line, but your hands exerted an opposing force to keep you on. The tendency for your body to fly off the merry-go-round is called centrifugal force. It isn’t a real force, but an apparent one. The force you used with your hands to stay on the ride is real, and it is called centripetal force. Let’s learn more about it.

### Centripetal Force

Centripetal force is a force on an object directed to the center of a circular path that keeps the object on the path. Its value is based on three factors: 1) the velocity of the object as it follows the circular path; 2) the object’s distance from the center of the path; and 3) the mass of the object.

Centrifugal force, on the other hand, is not a force, but a tendency for an object to leave the circular path and fly off in a straight line. Sometimes people mistakenly say ‘centrifugal force’ when they mean ‘centripetal force.’ The velocity of the object is constant and perpendicular to a line running from the object to the center of the circle; it is called tangential velocity.

## Force Centripetal Equation

Any motion in a curved path represents accelerated motion, and requires a force directed toward the center of curvature of the path. This force is called the centripetal force which means “center seeking” force. The force has the magnitude.

F(centriptal)= mv^2 / r

Swinging a mass on a string requires string tension, and the mass will travel off in a tangential straight line if the string breaks.

The centripetal acceleration can be derived for the case of circular motion since the curved path at any point can be extended to a circle.

## Centripetal Force Formula Questions

1) If a 150g ball is tied to a pole with a rope of length 1.5 m, and it spins around the pole at 30 m/s, what is the Centripedal Force?

Answer: The mass of the ball, m = 150 g = 0.150 kg. The radius, r = 1.5 m, and the velocity, v = 30 m/s.

Fc = mv2/ r

Fc = (0.150 kg)(30 m/s)2 / 1.5 m

Fc = 135 kgm2/s2/ 1.5 m

Fc = 90 kg m/s2 = 90 Newtons

2) Susan is holding on to the outer edge of a merry-go-round that is 1.8 m in diameter. Susan’s weight is 40 kg, and the velocity of the merry-go-round is 2.5 m/s. What is the centripedal force?

Answer: The radius, r = 1/2 d = 1/2 x 1.8 m = 0.9 m. The mass, m = 40 kg, and the velocity, v = 2.5 m/s.

Fc = mv2/ r

Fc = (40 kg)(2.5 m/s)2/ 0.9 m

Fc = 250 kg m2/s2/ 0.9 m

Fc = 277.77 Newtons

## 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:

a = v2 / r

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:

F = mv2 / r

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.

## 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:

F = Gm1m2 / r2

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 = 2π_r, 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. 