6. Moments of Inertia by Integration

by M. Bourne

The moment of inertia is a measure of the resistance of a rotating body to a change in motion.

The moment of inertia of a particle of mass m rotating about a particular point is given by:

`"Moment of inertia" = md^2`

where d is the radius of rotation.

Inertia for a Collection of Particles

If a group of particles with masses m1, m2, m3, ... , mn is rotating around a point with distances d1, d2, d3, ... dn, (respectively) from the point, then the moment of inertia I is given by:

I = m1d12 + m2d22 + m3d32 +... + mndn2

If we wish to place all the masses at the one point (R units from the point of rotation) then

d1 = d2 = d3 = ... = dn = R and we can write:

I = (m1 + m2 + m3 ... + mn)R2

R is called the radius of gyration.

Example 1

Find the moment of inertia and the radius of gyration w.r.t. the origin (0,0) of a system which has masses at the points given:

Mass `6` `5` `9` `2`
Point `(-3, 0)` `(-2, 0)` `(1, 0)` `(8, 0)`

Moment of Inertia for Areas

area between curves

We want to find the moment of inertia, Iy of the given area, which is rotating around the y-axis.

Each "typical" rectangle indicated has width dx and height y2y1, so its area is (y2y1)dx.

If k is the mass per unit area, then each typical rectangle has mass k(y2 y1)dx.

The moment of inertia for each typical rectangle is [k(y2 y1)dx] x2, since each rectangle is x units from the y-axis.

We can add the moments of inertia for all the typical rectangles making up the area using integration:

`I_y=kint_a^bx^2(y_2-y_1)dx`

Using a similar process that we used for the collection of particles above, the radius of gyration Ry is given by:

`R_y=sqrt((I_y)/m`

where m is the mass of the area.

Example 2

For the first quadrant area bounded by the curve

`y = 1 − x^2`,

find:

a) The moment of inertia w.r.t the y axis. (Iy)

b) The mass of the area

c) Hence, find the radius of gyration

Rotation about the x-axis

For rotation about the x-axis, the moment of inertia formulae become:

`I_x=kint_c^(d)y^2(x_2-x_1)dy`

and

`R_x=sqrt((I_x)/m`

Example 3

Find the moment of inertia and the radius of gyration for the area `y=x^2+1` from `x=1` to `x=2`, and `y>1`, when rotated around the x-axis. The mass per unit area is `3` kg m−2.