EXAMPLE 3: MASS MOMENT OF INERTIA

Calculate the mass moment of inertia of the parabolic rod about the y-axis. Assume the rod is made of a uniform material and has a mass of m.

Solution:

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

The length of the bar can be calculated from

The element of arc length in a rectangular coordinate system can be written as

The equation for the parabola is

Substitution of the point (a, h) into this equation givens the equation of the bar as

The length of the bar can, therefore, be calculated as

The distance from the y-axis is x. Therefore, r=x. The mass moment of inertia about the y-axis can be written as

For a uniform bar the density can be calculated using the total mass and total length of the bar so that