Calculus Proof of Centripetal Acceleration Magnitude

In basic circular motion in physics we are given the following equation for the centripetal acceleration on a body moving in circular motion:

However how can this be proved? Well, acceleration is a vector- so what we need to know is it’s magnitude and direction. The second part of which is obvious, it’s in the name “centripetal”. The direction of the acceleration vector will always be towards the centre of the circular path. So what we really need to prove is the magnitude of the acceleration vector. But how do we do this?

First lets take an object with a constant linear velocity in a circular path, and say it’s position at time t is the vector P(t) and the angle it makes with the positive x axis θ. With the origin being the centre of the circle the object is moving around:

Since the origin is the centre (we said so) the magnitude of the position vector will always be equal to the radius of the circle (even though the direction will change), this will help us later on:

As P(t) is a vector we can resolve it into it’s horizontal and vertical components using simple trig, for any time “t” along it’s path:

So now we have the position of the particle at any time we can do some simple calculus (chain rule) to find it’s velocity. We know that the velocity of the particle will be the derivative of the displacement vector so:

We now have a really annoying dθ/dt. But is it so annoying? NO! The rate of change of angle with respect to time is the angular velocity! We can now clean up our derivative.

We now have the velocity of the particle for any time “t” so how do we get to acceleration? You guessed it, another derivative:

We are going to assume that angular velocity is a constant (and can be treated as a scalar), so we can factor it out of our vector:

Do you recognize what is left in the vector? It’s our original position vector!! So we can therefore say:

Now we wanted to prove the magnitude of the centripetal acceleration so we can take the magnitude of both sides of our equation, the minus sign on the angular velocity vanishes, and the magnitude of the position vector we defined at the start finally comes in handy:

Almost done! Now we also know how to substitute for angular velocity for linear velocity (if you don’t know visit this circular motion lesson) so we can finally get acceleration in terms of v and r:

And voila! Wasn’t too bad was it 🙂

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