First slide

Rotational motion

Question

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

Moderate

Solution

$\begin{array}{l} C o n s e r v a t i o n o f A n g u l a r M o m e n t u m, \\ (\frac{M R^{2}}{2}) ω_{1} = (\frac{M R^{2}}{2} + \frac{M \frac{R^{2}}{4}}{2}) ω_{2} \\ \Rightarrow ω_{2} = \frac{4}{5} ω_{1} \\ G i v e n : e n e r g y l o s t = p % o f i n i t i a l e n e r g y \\ \Rightarrow (K_{i} - K_{f}) = \frac{p}{100} \times K_{i} \\ \Rightarrow p = \frac{(K_{i} - K_{f})}{K_{i}} \times 100 = \frac{\frac{1}{2} (\frac{M R^{2}}{2}) {ω_{1}}^{2} - \frac{1}{2} (\frac{M R^{2}}{2} + \frac{M \frac{R^{2}}{4}}{2}) {ω_{2}}^{2}}{\frac{1}{2} (\frac{M R^{2}}{2}) {ω_{1}}^{2}} \times 100 \\ \Rightarrow p = (1 - \frac{5}{4} {(\frac{ω_{2}}{ω_{1}})}^{2}) \times 100 \\ \Rightarrow p = (1 - \frac{4}{5}) \times 100 = 20 \end{array}$

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