Question

A wheel is rotating at an
angular speed w about its axis which is kept vertical. Another wheel of the same
radius but half the mass, initially at rest, is slipped on the same axle
gently. These two wheels then rotate with a common speed. Calculate the
common speed.

Answer 1

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Answer 2

Answer:

The common angular velocity is  $\boldsymbol{\frac{2}{3}\omega}$

Explanation:

Angular speed of the wheel = ω

If the mass of the wheel is M and radius R then the Moment of Inertia of the wheel

$I=MR^2$

When another wheel with mass M/2 is placed on the wheel then its Moment of Inertia = $\frac{M}{2}R^2$

The combined moment of inertia of the wheel of mass M and the wheel of mass M/2

$I'=MR^2+\frac{M}{2}R^2$

$\implies I'=\frac{3}{2}MR^2$

If the common angular velocity is ω' then from the conservation of Angular Momentum

$I\omega=I'\omega'$

$\implies MR^2\omega=\frac{3}{2}MR^2\omega'$

$\implies \omega'=\frac{2}{3}\omega$

Thus, the common angular velocity is $\frac{2}{3}\omega$

Hope it helps.