A thin circular disc of radius R has surface mass density σ(r) = σ₀r/R, where r is the distance from its geometrical centre. If the distance of its centre of mass from the geometrical centre is NR, find N.
Physics · Center of maas , Momentum and Collision · center of mass and question based on it
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Question

A thin circular disc of radius R has surface mass density σ(r) = σ₀r/R, where r is the distance from its geometrical centre. If the distance of its centre of mass from the geometrical centre is NR, find N.

Physics · Center of maas , Momentum and Collision · center of mass and question based on it

G1 Physics · Accepted Answer

Although the density changes with radial distance r, it has no angular dependence. Therefore every mass element at angle φ has an identical counterpart at angle φ+π. Their moments about the centre cancel pairwise. The mass distribution remains rotationally symmetric, so the centre of mass stays exactly at the geometrical centre. Hence the distance is 0 and N = 0.