**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

_{}