Question 218 of 280

A circular disc of mass $\text{M}$ and radius $\text{R}$ is rotating about its axis with angular speed ${\omega }_{1}$. If another stationary disc having radius $\dfrac{\text{R}}{2}$ and same mass $\text{M}$ is dropped co-axially on to the rotating disc. Gradually both discs attain constant angular speed ${\omega }_{2}$. The energy lost in the process is \text{p%} of the initial energy. Value of $p$ is ______

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Answer: 20

${I}_{1}{\omega }_{1}+{I}_{2}{\omega }_{2}=\left({I}_{1}+{I}_{2}\right)\times {\omega }_{f}$

$\dfrac{{MR}^{2}}{2}\times \omega +0\left(\dfrac{{MR}^{2}}{2}+\dfrac{{MR}^{2}}{8}\right)\dfrac{16}{25}$
${\omega }_{f}=\dfrac{4}{5}\omega$
Final $K.E$., ${K}_{f}=\dfrac{1}{2}\left(\dfrac{{MR}^{2}}{2}+\dfrac{{MR}^{2}}{8}\right)\dfrac{16}{25}{\omega }^{2}$
${K}_{f}=\dfrac{{MR}^{2}{\omega }^{2}}{5}$ ; ${K}_{i}=\dfrac{1}{2}\left(\dfrac{{MR}^{2}}{2}\right){\omega }^{2}=\dfrac{{MR}^{2}{\omega }^{2}}{4}$
Percentage loss in kinetic energy $$