Question

Use proper calculus. (a) A conical shell has mass M, height h, and base radius R, Assume it is made from thin sheet of uniform thickness, Derive its center of mass and moment of inertia about its symmetry axis. (b) Derive the formula for the center of mass and the moment of inertia of a solid sphere, and then that of the moment of inertia about an axis tangent to the surface. (c) Derive the formula for the moment of inertia of a thin shell (hollow sphere).

Answer 1

a)Let us consider a solid cone of mass M,base radius R and height h.The mass per unit volume of cone is given by,πR^2h.Let be the semivertical angle of cone .Let us consider a small circular disc of radius r at a distance from the vertex.Let the disc have a thickness x.Hence the volume of disc=πr^2dx.

Mass of disc=(3M/πR^2h)πr^2dx=3Mr^2dx/R^2h

M.I of disc about vertical axis perpendicular to its plane is given by,=1/2(3Mr^2dx/R^2h)r^2

=(3MR^2/2h^2)x^4dx

M.I of solid cone about its vertical axis is given by,

=

=3/10(MR^2)

b) Let us consider a solid sphere of radius R and mass M

M.I of disc about diameter is given by I=π/15.R^5

Mass of sphere is M=4/3πR^3

I=2(4/3πR^3)(R^2/)=2/5MR^2

C)M.I of a sphere about its central axis,I=MR^2,M=mass of sphere ,R=radius of sphere

For a hollow sphere,I=(2/5MR^2)-(2/5mr^2).