M and R be the mass and radius of disc. A small disc of radius R/3 is removed from the bigger dise as shown in figure. The moment of inertia of remaining part of bigger disc about an axis AB passing through the centre O and perpendicular to the plane of d

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Solution

Q.
M and R be the mass and radius of a disc. A small disc of radius R/3 is removed from the bigger dise as shown in figure. The moment of inertia of remaining part of bigger disc about an axis AB passing through the centre O and perpendicular to the plane of disc is $\frac{4}{x} M R^{2}$ . The value of x is __________. [2025]

Ans.
(9)

Without cavity $I_{1} = \frac{M R^{2}}{2}$

Mass of removed disc = $\frac{M}{π R^{2}} \times {(\frac{R}{3})}^{2} π = (\frac{M}{9})$

M.I. of removed disc $I_{2} = \frac{\frac{M}{9} {(\frac{R}{3})}^{2}}{2} + \frac{M}{9} \times {(\frac{2 R}{3})}^{2} = \frac{M R^{2}}{18}$

$I_{(remaining part)} = I_{1} - I_{2} = \frac{M R^{2}}{2} - \frac{M R^{2}}{18} = \frac{4 M R^{2}}{9} \Rightarrow x = 9$

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