A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density and radius 2R, as shown in the figure. The moment of inertia of this lamina about axes passing through O and P is $I_{O}$ and $I_{P}$ respectively. Both these axes are perpendicular to the plane of the lamina. The ratio $I_{P} / I_{O}$ to the nearest integer is [2012]

Question

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(3)

Let $σ$ be the surface mass density.

Moment of inertia of the lamina about axes passing through 'O'

$I_{O} = \frac{1}{2} σ [π {(2 R)}^{2}] \times {(2 R)}^{2} - [\frac{1}{2} {(σ π R^{2})}^{2} + σ (π R^{2}) \times R^{2}]$

$= \frac{13}{2} π σ R^{4}$

Moment of inertia of the lamina about axes passing through P

$I_{P} = 8 π σ R^{4} + σ π {(2 R)}^{2} \times {(2 R)}^{2} - [\frac{1}{2} σ (π R^{2}) R^{2} + σ (π R^{2}) {(\sqrt{{(2 R)}^{2} + R^{2}})}^{2}]$

$= 24 π σ R^{4} - 5.5 π σ R^{4} = 18.5 π σ R^{4}$

$∴$ $\frac{I_{P}}{I_{O}} = \frac{18.5 π σ R^{4}}{\frac{13}{2} π σ R^{4}} = \frac{37}{13} \approx 3$

Solution

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