First slide

Moment of intertia

Question

A hollow disc of inner radius R and outer radius 2R has moment of inertia I about an axis passing through its centre and perpendicular to the plane of the disc. When the disc is melted and recast into a solid disc of same thickness, its moment of inertia about the same axis is

Difficult

A

$\frac{4}{3} I$
B

$\frac{2}{5} I$
C

$\frac{3}{4} I$
D

$\frac{3}{5} I$

Solution

$\begin{array}{l} Mass (m) = π [{(2 R)}^{2} - R^{2}] t . ρ = π {(R^{1})}^{2} . t . ρ \\ \Rightarrow R^{1} = \sqrt{3} R \end{array}$
Now, $I = \frac{1}{2} m [{(2 R)}^{2} + R^{2}] = \frac{5}{2} {mR}^{2}$
$I^{1} = \frac{1}{2} m {(R^{1})}^{2} = \frac{1}{2} . m {(\sqrt{3} R)}^{2} = \frac{3}{2} {mR}^{2} = \frac{3}{2} . \frac{2 I}{5} = \frac{3}{5} I$

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