A rod of length ℓ has nonuniform linear mass density λ(x)=λ₀x, with x measured from one end. If centre of mass is kℓ/3 from that end, find k.
Physics · Mixed Chapter
JEE Main Practice

Question

A rod of length ℓ has nonuniform linear mass density λ(x)=λ₀x, with x measured from one end. If centre of mass is kℓ/3 from that end, find k.

G1 Physics · Accepted Answer

Step 1: For a continuously distributed mass, x_cm=∫x·λ(x)dx/∫λ(x)dx. Step 2: x_cm=[λ₀∫₀ˡ x²dx]/[λ₀∫₀ˡ xdx] = (ℓ³/3)/(ℓ²/2)=2ℓ/3. Step 3: Compare x_cm=kℓ/3; hence k=2.