A solid right circular cylinder of radius and height has its axis along the -axis

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Vector calculus and field geometry

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

Problem

A solid right circular cylinder of radius R=0.100 mR = 0.100\,\mathrm m and height h=0.400 mh = 0.400\,\mathrm m has its axis along the zz-axis. Its mass density depends only on the perpendicular distance ρ\rho from that axis, ρm(ρ)=ρm0(1−ρ/R)\rho_{\mathrm m}(\rho) = \rho_{\mathrm m0}(1 - \rho/R) with ρm0=1200 kg m−3\rho_{\mathrm m0} = 1200\,\mathrm{kg\,m^{-3}}, falling linearly from ρm0\rho_{\mathrm m0} on the axis to zero at the rim. ((a)) Choose coordinates for the mass integral, saying why they suit this solid, and compute the mass MM. ((b)) Compute the moment of inertia II about the central (zz) axis. ((c)) Evaluate I/(MR2)I/(MR^2) and compare it with the value 1/21/2 for a uniform cylinder, explaining the sign of the difference.

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