A uniform solid cylinder of length L and radius R has moment of inertia I₁ about its own axis. A small co-centric cylinder of length L/2 and radius R/3 is carved from this cylinder and has moment of inertia I₂ about the same axis. The ratio (I₁)/(I₂) is .

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Accepted Answer

162. Both pieces are the same material, so they share a density ρ. Large cylinder: M₁=ρπ R²L and I₁=1/2M₁R²=1/2ρπ R⁴L. Carved cylinder: M₂=ρπ(R/3)²L/2=(ρπ R²L)/(18). I₂=1/2M₂(R/3)²=1/2·(ρπ R²L)/(18)·(R²)/9=(ρπ R⁴L)/(324). (I₁)/(I₂)=(1/2ρπ R⁴L)/(1/(324)ρπ R⁴L)=(324)/2=162. Check the two factors separately: the length contributes a factor 2 and the radius a factor 3⁴=81, and 2×81=162.

A uniform solid cylinder of length L and radius R has moment of inertia I₁ about its own axis. A small co-centric cylinder of length L/2 and radius R/3 is carved from this cylinder and has moment of inertia I₂ about the same axis. The ratio (I₁)/(I₂) is ______.

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