Mathematical Methods for Physics and Engineering - Matematica.NET

MULTIPLE INTEGRALSaθdCFigure 6.8Suspending a semicircular lamina from one of its corners.6.3.4 Moments of inertiaFor problems in rotational mechanics it is often necessary to calculate the momentof inertia of a body about a given axis. This is defined by the multiple integral∫I = l 2 dM,where l is the distance of a mass element dM from the axis. We may again choosemass elements convenient for evaluating the integral. In this case, however, inaddition to elements of constant density we require all parts of each element tobe at approximately the same distance from the axis about which the moment ofinertia is required.◮ Find the moment of inertia of a uniform rectangular lamina of mass M with sides a andb about one of the sides of length b.Referring to figure 6.9, we wish to calculate the moment of inertia about the y-axis.We therefore divide the rectangular lamina into elemental strips parallel to the y-axis ofwidth dx. The mass of such a strip is dM = σb dx, whereσ is the mass per unit area ofthe lamina. The moment of inertia of a strip at a distance x from the y-axisissimplydI = x 2 dM = σbx 2 dx. The total moment of inertia of the lamina about the y-axis isthereforeI =∫ a0σbx 2 dx = σba33 .Since the total mass of the lamina is M = σab, wecanwriteI = 1 3 Ma2 . ◭198