A small mass m tied to a piece of thread moves over a smooth horizontal plane/ The other end of the thread is drawn through a hole with constant velocity. Show that the tension in the thread is inversely proportional to the cube of the distance from the hole.


Angular momentum of the mass is assumed to be constant. The particle velocities \(v\) and \(r\) are perpendicular. The angular momentum $$ J=mvr=constant$$
$$ v\propto 1/r$$
The tension T arises from centripetal force $$ T=\frac{mv^2}{r}
$$ Hence
$$ T\propto \frac{1}{r^2}\frac{1}{r}$$ or
$$ T\propto 1/r^3$$