Question

Discs $A$ and $B$ are mounted on a shaft $SS$ and may be connected or disconnected by a clutch $C$. The moment of inertia of disc $A$ is one half that of disc $B$. With the clutch disconnected, $A$ is brought up to an angular velocity ${\omega }_0$. The accelerating torque is then removed from $A$ and it is coupled to disc $B$ by the clutch (Bearing friction may be neglected). It is found that $2000J$ of heat is developed in the clutch when the connection is made. What is the original kinetic energy of disc $A$?

Answer 1

Initial angular momentum$=I{\omega }_0$

$[I_A=I{I}_B=2I]$-Given

Final angular momentum$=3I\omega$

$\therefore$According to conservation of angular momentum

$I{\omega }_0=3I\omega$

Or $\omega =\frac{{\omega }_0}{3}$

Initial K.E.$=\frac{1}{2}I{\omega }_0^2=E$

Final K.E.$=\frac{1}{2}\left(3I\right){\omega }_0^2=\frac{1}{2}\left(3I\right)\frac{{\omega }_0^2}{9}=\frac{1}{2}I{\omega }_0^2\times \frac{1}{3}=\frac{1}{3}$

Energy lost as heat$=E-\frac{E}{3}=\frac{2E}{3}=2000J$

$\Rightarrow E=3000J$