# 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?

Feb 19, 2016

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

#### Explanation:

Since the kinetic energy is ${E}_{k} = \frac{1}{2} m {v}^{2}$ and increased from $2 J \to 16 J$, ie increased $8$ times, it implies that the ${v}^{2}$ must have increased $8$ times, and hence $v$ increased $\sqrt{8}$ times.
(Assuming mass remains constant.)

Now the centripetal force is given my ${F}_{c} = \frac{m {v}^{2}}{r}$ 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 $\sqrt{{E}_{k} / \left(\frac{1}{2} m\right)} = \sqrt{\frac{2}{\frac{1}{2} \times 2}} = \sqrt{2} m / s$

So initial centripetal force was $F c = \frac{2 \times 2}{4} = 1 N$.

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