A thin disc of radius R, centred at the origin, has surface mass density σ(x, y) = σ₀(1 + x/R). Its centre of mass lies at:
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A thin disc of radius R, centred at the origin, has surface mass density σ(x,y) = σ₀(1 + x/R). Its centre of mass lies at:

G1 Physics · Accepted Answer

The y-coordinate of the centre of mass is zero by reflection symmetry about the x-axis. Total mass M = ∫σ dA = σ₀∫dA + (σ₀/R)∫x dA = σ₀πR², because ∫x dA = 0 over a complete disc. Now ∫xσ dA = σ₀∫x dA + (σ₀/R)∫x²dA = (σ₀/R)(πR⁴/4). Therefore x(CM) = [(σ₀/R)(πR⁴/4)]/(σ₀πR²) = R/4.