Two iron solid discs of negligible thickness have radii and and moment of inertia and , respectively. For , the ratio of and would be 1/x, where x = __________. [2025]

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Solution

Q.
Two iron solid discs of negligible thickness have radii $R_{1}$ and $R_{2}$ and moment of inertia $I_{1}$ and $I_{2}$ , respectively. For $R_{2} = 2 R_{1}$ , the ratio of $I_{1}$ and $I_{2}$ would be 1/x, where x = __________. [2025]

Ans.
(16)

Given, $R_{2} = 2 R_{1}$

$M_{1} = σ \times π R_{1}^{2} = M_{o}$

$M_{2} = σ \times π R_{2}^{2} = σ \times π {[2 R_{1}]}^{2} = σ \times 4 π R_{1}^{2} = 4 M_{o}$

$\frac{I_{1}}{I_{2}} = \frac{\frac{M_{1} R_{1}^{2}}{2}}{\frac{M_{2} R_{2}^{2}}{2}} = \frac{M_{1} R_{1}^{2}}{M_{2} R_{2}^{2}} = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$

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