Let’s discuss a problem on angular momentum and prove that the tension in the thread is inversely proportional to the cube of the distance from the hole.

**The Problem:**

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.

**Discussion:**

Angular momentum of the mass is assumed to be constant. The particle velocities (v) and (r) are perpendicular. The angular momentum $$ J=mvr=constant$$

Hence,

$$ v\propto 1/r$$

The tension T arises from centripetal force $$ T=\frac{mv^2}{r}

$$ Hence

$$ T\propto \frac{1}{r^2}\frac{1}{r}$$ or

$$ T\propto 1/r^3$$

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