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 throu

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 0 ​ and I p ​ , respectively. Both these axes are perpendicular to the plane of the lamina. The ratio I o ​ I p ​ ​ to the nearest integer is :

TestHub · Accepted Answer

Let the removed disc have mass M. The original disc, with twice its radius, has mass 4M.

Distances of the removed centre from O and P are
$R, 5 R$

$I_{O, full} = 8 M R^{2}$

$I_{O, removed} = \frac{3}{2} M R^{2}$

$I_{O} = (8 - \frac{3}{2}) M R^{2}$

$I_{O} = \frac{13}{2} M R^{2}$

$I_{P, full} = 24 M R^{2}$

$I_{P, removed} = \frac{1}{2} M R^{2} + 5 M R^{2}$

$I_{P, removed} = \frac{11}{2} M R^{2}$

$I_{P} = (24 - \frac{11}{2}) M R^{2}$

$I_{P} = \frac{37}{2} M R^{2}$

$\frac{I _{P}}{I _{O}} = \frac{37}{13}$

$\frac{I _{P}}{I _{O}} ≃ 2.85$

The nearest integer is 3.

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