A uniform wire forms a quarter-circle arc of radius R, extending from θ = 0 to θ = π/2. Its centre of mass has coordinates x(CM) = y(CM) = NR/π. Find N.

Question

G1 Physics · Accepted Answer

For a uniform arc, dm = λR dθ and total length is R(π/2), so total mass is λRπ/2. For the x-coordinate, x(CM) = [∫₀^(π/2) (R cosθ)(λR dθ)]/[λRπ/2] = [λR²(1)]/[λRπ/2] = 2R/π. Similarly, y(CM) = 2R/π. Comparing with NR/π gives N = 2.