A uniform solid cylinder of length and radius has moment of inertia about its axis equal to . A small co-centric cylinder of length and radius carved from this cylinder has moment of inertia about its axis equals to . The ratio is _________.

Question

A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_{1}$ . A small co-centric cylinder of length $\frac{L}{2}$ and radius $\frac{R}{3}$ carved from this cylinder has moment of inertia about its axis equals to $I_{2}$ . The ratio $I_{1} / I_{2}$ is _________. [2026]

Smart Achievers · Accepted Answer

(162)

$Original mass (M)$

$The removed mass (m)$

$m = ρ \times π {(\frac{R}{3})}^{2} \times \frac{L}{2}$

$= \frac{ρ . π R^{2} L}{18} = \frac{M}{18}$

$I' = \frac{1}{2} \cdot \frac{M}{18} \cdot \frac{R^{2}}{9} = \frac{1}{324} M R^{2}$

$\frac{I}{I'} = \frac{\frac{1}{2} M R^{2}}{\frac{1}{324} M R^{2}} = 162$

Solution

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Solution

Q. A uniform solid cylinder of length L and radius R has moment of inertia about its axis equal to I1. A small co-centric cylinder of length L2 and radius R3 carved from this cylinder has moment of inertia about its axis equals to I2. The ratio I1/I2 is _________. [2026]

Ans. (162) Original mass (M) The removed mass (m) m=ρ×π(R3)2×L2 =ρ.πR2L18=M18 I'=12·M18·R29=1324MR2 II'=12MR21324MR2=162

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