Question

Given the mass and the velocity of an object in circular motion with radius `r`, how do we calculate the magnitude of the centripetal force, `F`?

Answer 1

Constant acceleration (change of direction) requires a constant force, which is described by `F=(mv^2)/r` if `v` is measured in `ms^-1` or `F=momega^2r` if `omega` is measured in `rads^-1`.

Explanation:

Acceleration is defined as the rate of change of velocity, and velocity has a direction.

Since the direction of something in circular motion is always changing, it requires a constant acceleration, and Newton's Second Law indicates that constant acceleration requires a constant force.

`F=ma`

If we look at the instantaneous linear velocity of the object at any moment as `v` `ms^-1`, the expression for the centripetal acceleration is:

`a=v^2/r`

So the force acting is simply `F=(mv^2)/r`

Let's look at where that expression for the acceleration comes from.

The speed of the object is constant: it is not accelerating because its speed increases (speeding up) or decreases (slowing down). It is accelerating because its direction is changing constantly.

If we imagine a direction vector, which is a tangent to the circle in which the object is moving, at each moment it points in a slightly different direction.

Acceleration is just change in velocity divided by change in time:

`a=(Delta v)/(Delta t)`

This Khan Academy video offers a more-detailed explanation of how we get from there to `a=v^2/r`: https://www.khanacademy.org/science/physics/centripetal-force-and-gravitation/centripetal-acceleration-tutoria/v/visual-understanding-of-centripetal-acceleration-formula

If, instead, we measure the radial velocity of the object in radians per second `(rads^-1)`, the expression is:

`F=m omega^2 r`