A circular plate of radius has surface mass density

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Calculus of change and accumulation

Mathematical Methods for Physics, Ch Multiple integrals and change of variables

Problem

A circular plate of radius R=0.30 mR = 0.30\,\mathrm{m} has surface mass density σ(r)=σ0(r/R)n\sigma(r) = \sigma_0(r/R)^n, where rr is the distance from the centre, σ0=1.5 kg m−2\sigma_0 = 1.5\,\mathrm{kg\,m^{-2}}, and n≥0n \ge 0 is a fixed power. (a) Show that the mass is M(n)=2πσ0R2n+2M(n) = \dfrac{2\pi\sigma_0 R^2}{n+2}. (b) Show that the moment of inertia about the axis through the centre perpendicular to the plate is I(n)=2πσ0R4n+4I(n) = \dfrac{2\pi\sigma_0 R^4}{n+4}. (c) Evaluate MM, II and the ratio I/(MR2)I/(MR^2) for n=1n=1 and for n=3n=3, and say how the ratio changes as mass is pushed further towards the rim.

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