wo uniform circular discs are rotating independently in the same direction around their common axis passing through their centres.
Question:

Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are $0.1 \mathrm{~kg}-\mathrm{m}^{2}$ and $10 \mathrm{rad} \mathrm{s}^{-1}$ respectively while those for the second one are $0.2 \mathrm{~kg}-\mathrm{m}^{2}$ and $5 \mathrm{rad} \mathrm{s}^{-1}$ respectively. At some instant they get stuck together and start rotating as a single system about their common axis with some angular speed. The Kinetic energy of the combined system is :

1. $\frac{10}{3} \mathrm{~J}$

2. $\frac{2}{3} \mathrm{~J}$

3. $\frac{5}{3} \mathrm{~J}$

4. $\frac{20}{3} \mathrm{~J}$

Correct Option:

Solution:

– Both discs are rotating in same sense

– Angular momentum conserved for the system

i.e. $\quad \mathrm{L}_{1}+\mathrm{L}_{2}=\mathrm{L}_{\text {final }}$

$\mathrm{I}_{1} \omega_{1}+\mathrm{I}_{2} \omega_{2}=\left(\mathrm{I}_{1}+\mathrm{I}_{2}\right) \omega_{\mathrm{f}}$

$0.1 \times 10+0.2 \times 5=(0.1+0.2) \times \omega_{\mathrm{f}}$

$\omega_{\mathrm{f}}=\frac{20}{3}$

– Kinetic energy of combined disc system

$\Rightarrow \frac{1}{2}\left(\mathrm{I}_{1}+\mathrm{I}_{2}\right) \omega_{\mathrm{f}}^{2}$

$=\frac{1}{2}(0.1+0.2) \cdot\left(\frac{20}{3}\right)^{2}$

$=\frac{0.3}{2} \times \frac{400}{9}=\frac{120}{18}=\frac{20}{3} \mathrm{~J}$