A rod of length ℓ has linear mass density λ = ax, where x is measured from the left end. What is its moment of inertia about an axis through its centre of mass and perpendicular to the rod?
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Question

A rod of length ℓ has linear mass density λ = ax, where x is measured from the left end. What is its moment of inertia about an axis through its centre of mass and perpendicular to the rod?

G1 Physics · Accepted Answer

Here dm=ax dx. Total mass M=∫₀^ℓ ax dx=aℓ²/2 and x of CM=[∫₀^ℓ x·ax dx]/M=2ℓ/3. Thus I about CM=∫₀^ℓ (x-2ℓ/3)² ax dx=aℓ⁴/36. Hence option B.