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Correlation to Linear Motion

Two rigid bod...

Question

Two rigid bodies $A$ and $B$ rotate with angular momenta $L_{A}$ and $L_{B}$ respectively. The moments of inertia of $A$ and $B$ about the axes of rotation are $I_{A}$ and $I_{B}$ respectively. If $I_{A} = I_{B} / 4$ and $L_{A} = 5 L_{B}$ . Then the ratio of rotational kinetic energy $K_{A}$ of $A$ to the rotational energy $K_{B}$ of $B$ is given by

A

\frac{K_{A}}{K_{B}} = \frac{25}{4}

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B

\frac{K_{A}}{K_{B}} = \frac{5}{4}

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C

\frac{K_{A}}{K_{B}} = \frac{1}{4}

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D

\frac{K_{A}}{K_{B}} = 100

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Solution

The correct option is D $\frac{K_{A}}{K_{B}} = 100$
Rotational K.E, $K = \frac{1}{2} I w^{2}$
Also, $L = I w ⟹ K = \frac{L^{2}}{2 I}$
$\frac{K_{A}}{K_{B}} = \frac{L_{A}^{2}}{L_{B}^{2}} \frac{I_{B}}{I_{A}}$
$\frac{K_{A}}{K_{B}} = \frac{25 L_{B}^{2}}{L_{B}^{2}} \frac{4 I_{A}}{I_{A}}$
$\frac{K_{A}}{K_{B}} = 100$

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