Single Choice

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$$\cfrac { { K }_{ A } }{ { K }_{ B } } =\cfrac { 25 }{ 4 } $$
B$$\cfrac { { K }_{ A } }{ { K }_{ B } } =\cfrac { 5 }{ 4 } $$
C$$\cfrac { { K }_{ A } }{ { K }_{ B } } =\cfrac { 1 }{ 4 } $$
D$$\cfrac { { K }_{ A } }{ { K }_{ B } }=100$$
Correct Answer

Solution

Rotational K.E, $$K= \dfrac{1}{2}I w^2$$
Also, $$L= Iw \implies K= \dfrac{L^2}{2I}$$
$$\dfrac{K_A}{K_B}= \dfrac{L_A^2}{L_B^2} \dfrac{I_B}{I_A}$$
$$\dfrac{K_A}{K_B}= \dfrac{25L_B^2}{L_B^2} \dfrac{4I_A}{I_A}$$
$$\dfrac{K_A}{K_B}= 100 $$

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