# the uniform circular discs are rotating in the same direction around their common axis passing through their centers. The  moment of inertia and angular velocity of the first disc are $0.1 \ kgm^2$ and $10 \ rad. s^{-1}$  respectively, while those for the second one are $0.2 \ kg.m^2$ and $5 \ rad. s^{-1}$ 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  Option: 1 Option: 2 Option: 3 Option: 4

$\\ \text{By angular momentum conservation}\\ I_1 \omega_1+I_2 \omega_2=(I_1+I_2) \omega\\ 0.1 \times 10+ 0.2 \times 5=(0.3) \times \omega\\ \omega=\frac{20}{3}\\ K.E.=\frac{1}{2}I \omega^2=\frac{1}{2}(0.1+0.2) \left[ \frac{20}{3}\right ]^2=\frac{20}{3}J$

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