Question

Two discs $A$ and $B$ having moment of inertia $I$ and $2I$ respectively, rotating initially with angular velocity $2\omega$ and $\omega$ are placed together coaxially on a vertical axle as shown in figure. Find the loss in kinetic energy of combination of discs, considering interaction between discs to be gentle.

• A. $\frac{I{\omega }^2}{2}$
• B. $\frac{I{\omega }^2}{3}$
• C. $\frac{I{\omega }^2}{4}$
• D. $\frac{I{\omega }^2}{6}$

Answer 1

The correct option is B $\frac{I{\omega }^2}{3}$

Let both discs rotate together with common angular velocity ${\omega }_0$
Moment of inertia of combination of disc about same axis of rotation,
$I_0=I+2I$
$I_0=3I$
$\Rightarrow {\tau }_{\text{ext}}=0$, on system of discs as force acting between them becomes internal to system and ${\tau }_{\text{gravity}}=0$

$L_i=L_f$
$\Rightarrow I\left(2\omega \right)+2I\left(\omega \right)=3I{\omega }_0$
$\Rightarrow 4I\omega =3I{\omega }_0$
$\therefore {\omega }_0=\frac{4\omega }{3}$
System has only $K{E}_{\text{Rot}}$ hence,
$\Rightarrow \Delta KE=K{E}_f-K{E}_i$
$\Rightarrow \Delta KE=\frac{1}{2}{I}_0{\omega }_0^2-(\frac{1}{2}{I}_A{\omega }_A^2+\frac{1}{2}{I}_B{\omega }_B^2)$
Or, $\Delta KE=\frac{1}{2}\times 3I\times {\left(\frac{4\omega }{3}\right)}^2-(\frac{1}{2}\times I\times (2\omega {)}^2+\frac{1}{2}\times 2I\times \left(\omega {)}^2\right)$
$\Delta KE=\frac{8I{\omega }^2}{3}-3I{\omega }^2$
$\therefore \Delta KE=\frac{-I{\omega }^2}{3}$
Here $-ve$ sign represents loss in kinetic energy for the system of discs.