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Q.

A circular disc of radius ‘ $r_{1}$ ’ and thickness $r_{1}$ /6 has moment of intertia ‘ $I$ ’ about an axis passing through is centre and perpendicular to its plane. It is melted and recasted into a solid sphere. The M.I of the sphere about its diameter as axis of rotation is

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a

$I / 10$

b

$I$

c

$I / 5$

d

$2 I / 3$

answer is C.

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Detailed Solution

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Volume disc = volume of sphere

$\begin{array}{l} π {r_{1}}^{2} \times t = \frac{4}{3} π {r_{2}}^{3}, π {r_{1}}^{2} \times \frac{r_{1}}{6} = \frac{4}{3} π {r_{2}}^{3}, {[\frac{r_{2}}{r_{1}}]}^{3} = \frac{1}{8}, r_{2} = \frac{r_{1}}{2} \end{array}$

$\begin{array}{l} I_{sp} = \frac{2}{5} m {r_{2}}^{2} = \frac{2}{5} m {(\frac{r_{1}}{2})}^{2} = \frac{2}{5} \frac{m {r_{1}}^{2}}{4} = \frac{1}{5} \frac{m {r_{1}}^{2}}{2} = \frac{I}{5} \end{array}$

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