A spherical cavity of radius $\frac{R}{2}$ is removed from a solid sphere of radius R. The sphere is placed on a rough horizontal surface as shown in the figure. The sphere is given a very little push. Friction is large enough to prevent slippage.

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The time period for very small oscillations is $T = 2 π \sqrt{\frac{32 R}{5 g}}$

The time period for very small oscillations is $T = 2 π \sqrt{\frac{177 R}{10 g}}$

The COM of the body is at a distance of $\frac{R}{21}$ from its centre.

The COM of the body is at a distance of $\frac{R}{7}$ from its centre.

answer is C.

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Detailed Solution

$x_{c m} = \frac{R}{14}$

Moment of inertial of cavitied sphere about an axis parallel to the plane through point of contact is
$\frac{4}{3} π R^{3} (1 - \frac{1}{8}) Ρ = M Ρ = \frac{3 M}{4 π R^{3}} \times (\frac{8}{7}) I_{P} = \frac{177}{140} M R^{2}$
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$M g \times \sin θ = M g (\frac{R}{14}) θ \frac{M g R}{14} θ = I_{α} = \frac{177}{140} M R^{2} α ω^{2} = \frac{10}{177} . \frac{g}{R} T = 2 π \sqrt{\frac{177 R}{10 g}}$