# A model train, with a mass of 4 kg, is moving on a circular track with a radius of 3 m. If the train's kinetic energy changes from 12 J to 48 J, by how much will the centripetal force applied by the tracks change by?

Mar 6, 2016

Centripetal force changes from $8 N$ to $32 N$

#### Explanation:

Kinetic energy $K$ of an object with mass $m$ moving at a velocity of $v$ is given by $\frac{1}{2} m {v}^{2}$. When Kinetic energy increases $\frac{48}{12} = 4$ times, velocity is hence doubled.

The initial velocity will be given by $v = \sqrt{2 \frac{K}{m}} = \sqrt{2 \times \frac{12}{4}} = \sqrt{6}$ and it will become $2 \sqrt{6}$ after increase in kinetic energy.

When an object moves in a circular path at a constant speed, it experiences a centripetal force is given by $F = m {v}^{2} / r$, where: $F$ is centripetal force, $m$ is mass, $v$ is velocity and $r$ is radius of circular path. As there is no change in mass and radius and centripetal force is also proportional to square of velocity,

Centripetal force at the beginning will be $4 \times {\left(\sqrt{6}\right)}^{2} / 3$ or $8 N$ and this becomes $4 \times {\left(2 \sqrt{6}\right)}^{2} / 3$ or $32 N$.

Hence centripetal force changes from $8 N$ to $32 N$