A uniform solid cylinder with radius R and length L has moment of inertia , about the axis of the cylinder. A concentric solid cylinder of radius and length is carved out of the original cylinder. If is the moment of inertia of the carved out portion o

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Solution

Q.
A uniform solid cylinder with radius R and length L has moment of inertia $I_{1}$ , about the axis of the cylinder. A concentric solid cylinder of radius $R' = \frac{R}{2}$ and length $L' = \frac{L}{2}$ is carved out of the original cylinder. If $I_{2}$ is the moment of inertia of the carved out portion of the cylinder, then $\frac{I_{1}}{I_{2}} =$ _______ . [2023]

Ans.
(32)

$I_{1} = \frac{m_{1} R^{2}}{2} I_{2} = \frac{m_{2} {(R / 2)}^{2}}{2}$

$\frac{I_{1}}{I_{2}} = \frac{4 m_{1}}{m_{2}} = \frac{4 \cdot ρ π R^{2} ℓ}{ρ \cdot \frac{π R^{2}}{4} \times \frac{ℓ}{2}}$ $\Rightarrow \frac{I_{1}}{I_{2}} = 32$

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