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.
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:
|Point||`(-3, 0)`||`(-2, 0)`||`(1, 0)`||`(8, 0)`|
Moment of Inertia for Areas
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 y2 − y1, so its area is (y2 − y1)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:
Using a similar process that we used for the collection of particles above, the radius of gyration Ry is given by:
where m is the mass of the area.
For the first quadrant area bounded by the curve
`y = 1 − x^2`,
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:
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.