Question 522: Centre of Mass and Rotational Motion

Question

Two bodies A and B have their moments of inertia $I_1$ and $I_2$ respectively about their axis of rotation. If their kinetic energies of rotation are equal and their angular momenta $\mathrm{L}_1$ and $\mathrm{L}_2$ respectively are in the ratio $1: \sqrt{3}$, then $I_2$ will be

Accepted Answer

$$\begin{aligned}
& 	ext { (K.E. })_{\mathrm{A}}=(	ext { K.E. })_{\mathrm{B}} \
& \frac{1}{2} \mathrm{I}_1 \omega_1^2=\frac{1}{2} \mathrm{I}_2 \omega_2^2 \
	herefore \quad & \frac{\omega_2^2}{\omega_1^2}=\frac{I_1}{I_2}\quad	ext{.... (i)}
\end{aligned}$$
$$
\begin{array}{ll} 
& 	ext { Also, K.E }=\frac{1}{2} \mathrm{~L} \omega \
	herefore & \frac{1}{2} \mathrm{~L}_1 \omega_1=\frac{1}{2} \mathrm{~L}_2 \omega_2 \
	herefore & \frac{\mathrm{~L}_1}{\mathrm{~L}_2}=\frac{\omega_2}{\omega_1}=\frac{1}{\sqrt{3}} \quad	ext{.... (ii)}\
	herefore & \frac{\mathrm{I}_1}{\mathrm{I}_2}=\frac{1}{3} \quad	ext{... [From (i) and (ii)]}\
	herefore & \mathrm{I}_2=3 \mathrm{I}_1
\end{array}$$

Answer

$\frac{1}{3} \mathrm{I}_1$

Answer

$\sqrt{3} I_1$

Answer

$2 \mathrm{I}_1$

Answer

$\mathrm{3 I_1}$

Centre of Mass and Rotational Motion

Two bodies A and B have their moments of inertia I_1 and I_2 respectively about their axis of rotation. If their kinetic energies of rotation are equal and their angular momenta \mathrm{L}_1 and \mathrm{L}_2 respectively are in the ratio 1: \sqrt{3}, then I_2 will be

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