Question

A model train, with a mass of `2 kg`, is moving on a circular track with a radius of `4 m`. If the train's kinetic energy changes from `2 j` to `16 j`, by how much will the centripetal force applied by the tracks change by?

Answer 1

It will increase `8` times, which in this case corresponds to an increase of `8N`.

Explanation:

Since the kinetic energy is `E_k=1/2mv^2` and increased from `2J to 16J`, ie increased `8` times, it implies that the `v^2` must have increased `8` times, and hence `v` increased `sqrt8` times.
(Assuming mass remains constant.)

Now the centripetal force is given my `F_c=(mv^2)/r` and directed towards the centre of the circle.
So assuming the mass and radius stays the same, if `v^2` increases `8` times, then `F_c` also increases `8 times`.

Now the initial velocity was `sqrt(E_k/(1/2m))=sqrt(2/(1/2xx2))=sqrt2 m//s`

So initial centripetal force was `Fc=(2xx2)/4=1N`.

Therefore the new centripetal force will be `8` times more, ie `8N`.