A rod has length ℓ and linearly varying mass density λ(x)=λ₀x. If centre-of-mass position is kℓ/3 from x=0, find k.
Physics · Mixed Chapter
JEE Main Practice

Question

A rod has length ℓ and linearly varying mass density λ(x)=λ₀x. If centre-of-mass position is kℓ/3 from x=0, find k.

G1 Physics · Accepted Answer

Step 1: Centre of mass xcm=∫₀ˡ xλ(x)dx / ∫₀ˡ λ(x)dx. Step 2: Numerator λ₀∫₀ˡ x²dx=λ₀ℓ³/3; denominator λ₀∫₀ˡ xdx=λ₀ℓ²/2. Step 3: xcm=(ℓ³/3)/(ℓ²/2)=2ℓ/3; k=2.