ABCDECA is a planar body of mass m of uniform thickness and same material. The dimensions are as shown in the figure. The moment of inertia of the body about an axis passing through point A and perpendicular to planar body is $I_{1}$ and that of about an axis passing through C and perpendicular to planer body is $I_{2} . Then \frac{I_{1}}{I_{2}}$ is___

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answer is 3.

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Detailed Solution

For $I_{C}$

$I_{x} = 2 (\frac{m}{2} \frac{a^{2}}{6}) = \frac{m a^{2}}{6},$ $I_{y} = 2 (\frac{m}{2} \frac{a^{2}}{6} + \frac{m}{2} {(\frac{2 a}{3})}^{2} - \frac{m}{2} {(\frac{a}{3})}^{2}) = \frac{m a^{2}}{2}$

$I_{C} = \frac{2}{3} m a^{2}$

For $I_{A}$

$I_{x} = \frac{m a^{2}}{26} + \frac{m}{2} \frac{a^{2}}{6} - \frac{m}{2} {(\frac{a}{3})}^{2} + \frac{m}{2} {(\frac{5 a}{3})}^{2} = \frac{5 m a^{2}}{6}$

$I_{y} = \frac{7 m a^{2}}{6}, I_{A} = 2 m a^{2}, ∴ \frac{I_{1}}{I_{2}} = 3$

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