Question

A model train with a mass of `4 kg` is moving along a track at `18 (cm)/s`. If the curvature of the track changes from a radius of `25 cm` to `42 cm`, by how much must the centripetal force applied by the tracks change?

Answer 1

The centripetal force changes in a factor of `25/42`, i.e. approximately `0.6` times greater.

Explanation:

The centripetal force acting on a moving mass `m` traveling a circular path with radius `r` at a constant speed `v` is given by the formula:

`F_c = m v^2/r`

If the path's radius is modified from a `r_1` value to a `r_2` one, the initial centripetal force `F_{c1}` changes to a new value `F_{c2}` which can be compared using the above formula:

`{F_{c2}}/{F_{c1}} = {m v^2/r_2}/{m v^2/r_1} = {1/r_2}/{1/r_1}=r_1 / r_2`

Thus:

`{F_{c2}}/{F_{c1}} = r_1 / r_2 = {25 cm} / {42 cm} ~~0.595... rArr F_{c2} ~~0.595 F_{c1}`