A solid hemisphere of steel

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 hemisphere of steel, treated as uniform, has radius R=0.150 mR = 0.150\,\mathrm m and density ρm=7800 kg m−3\rho_{\mathrm m} = 7800\,\mathrm{kg\,m^{-3}}. It occupies x2+y2+z2≤R2x^2+y^2+z^2 \le R^2, z≥0z \ge 0, with its flat circular face resting on the plane z=0z = 0. ((a)) Find its mass, and check the value by an independent calculation in a second coordinate system. ((b)) Find the moment of inertia II about its axis of symmetry, the zz axis. ((c)) Compare I/(MR2)I/(MR^2) with the value 2/52/5 for a full uniform sphere about a diameter, and explain what you find.

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