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Angular Momentum

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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