Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MULTIPLE INTEGRALS

a θ

d

C

Figure 6.8 Suspending a semicircular lamina from one of its corners.

6.3.4 Moments of inertia

For problems in rotational mechanics it is often necessary to calculate the moment

of inertia of a body about a given axis. This is defined by the multiple integral

I=

∫

l^2 dM,

wherelis the distance of a mass elementdMfrom the axis. We may again choose

mass elements convenient for evaluating the integral. In this case, however, in

addition to elements of constant density we require all parts of each element to

be at approximately the same distance from the axis about which the moment of

inertia is required.

Find the moment of inertia of a uniform rectangular lamina of massMwith sidesaand

babout one of the sides of lengthb.

Referring to figure 6.9, we wish to calculate the moment of inertia about they-axis.
We therefore divide the rectangular lamina into elemental strips parallel to they-axis of
widthdx. The mass of such a strip isdM=σb dx,whereσis the mass per unit area of
the lamina. The moment of inertia of a strip at a distancexfrom they-axisissimply
dI=x^2 dM=σbx^2 dx. The total moment of inertia of the lamina about they-axis is
therefore

I=

∫a

0

σbx^2 dx=

σba^3

3

.

Since the total mass of the lamina isM=σab,wecanwriteI=^13 Ma^2 .