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 IO and IP, respectively. Both these axes are perpendicular to the plane of the lamina. The ratio to the nearest integer is

Engineering

Physics

Moment of Inertia

Question

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 $\frac{I_{P}}{I_{O}}$ to the nearest integer is

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$I_{0} = \frac{1}{2} σπ {(2 R)}^{2} {(2 R)}^{2} - [\frac{1}{2} {σπR}^{2} {πR}^{2} + {σπR}^{2} R^{2}] = {σπR}^{4} [8 - \frac{3}{2}] = \frac{13}{2} {σπR}^{4}$

$I_{P} = \frac{3}{2} σπ {(2 R)}^{2} {(2 R)}^{2} - [\frac{1}{2} {σπR}^{2} {πR}^{2} + {σπR}^{2} 5 R^{2}]$

${σπR}^{4} [24 - \frac{11}{2}] = \frac{37 {σπR}^{4}}{2} ∴ \frac{I_{P}}{I_{O}} \underline{~} 3$