A grain silo is a cylinder of radius and height standing on a flat floor
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 grain silo is a cylinder of radius a=3.0m and height H=10m standing on a flat floor, closed at the top by a hemispherical dome of the same radius a, whose flat face coincides with the top of the cylinder. The silo is treated as a geometrically ideal cylinder-plus-hemisphere with no wall thickness, and the volume of grain it can hold equals its geometric interior volume.
((a)) Can any one of the Cartesian, cylindrical and spherical systems make every boundary of the silo (floor, wall and dome) a coordinate surface? If not, divide the silo into pieces for each of which one system does, and name the coordinate surface that each boundary of each piece becomes.
((b)) Write the volume element for each piece from its scale factors, and integrate it over the appropriate ranges to find the volume of each piece.
((c)) Add the volumes to give the total capacity of the silo.