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 Curvilinear and orthogonal coordinates
Problem
A spherical water tank has inner radius R=1.5m. Take the origin at its centre and the z axis vertically upward.
((a)) When the tank is exactly half full, the water is bounded by the curved wall and by its flat surface through the centre. Show that one of the three systems makes both boundaries coordinate surfaces, and find the volume of water by integrating that system's volume element.
((b)) The tank is then filled to a depth of 2.25m, so that the water surface is the plane z=0.75m. Explain why none of the three systems now makes every boundary of the water a coordinate surface.
((c)) Choose a system anyway, set up the volume integral with whatever limit your choice forces to vary, and find the volume of water. Say which boundary you allowed to give the varying limit, and why that choice is the easier one.