EXAMPLE 1: MASS MOMENT OF INERTIA

Calculate the mass moment of inertia of the cone about the z-axis. Assume the cone is made of a uniform material of density (mass per unit volume).

Solution:

The mass moment of inertia about the z-axis is given by

The element of volume in a cylindrical coordinate system is given by

The domain of the cone in cylindrical coordinates is defined by

Therefore, the mass moment of inertia about the z-axis can be written as

For a uniform cone the density can be calculated using the total mass and total volume of the cone so that

Therefore, the moment of inertia in terms of the total mass of the cone can be written as