A cylinder with radius ?R? and mass ?M? has density that increases linearly with distance ?r? from the cylinder axis, ??? = ??r?, where ??? is a positive constant. (a) Calculate the moment of inertia of the cylinder about a longitudinal axis through its center in terms of ?M? and ?R?. (b) Is your answer greater or smaller than the moment of inertia of a cylinder of the same mass and radius but of uniform density? Explain why this result makes qualitative sense.
Solution 97P Introduction We have to calculate the moment of inertia of the cylinder given the density of the cylinder varies linearly with the distance from the center. Then we have to find out if the result is less or higher than the case when the density is uniform. And discuss why. Step 1 The above figure showing the cross section of the cylinder. Consider that the thickness of the cylinder is h. The density of the cylinder is given by (r) = r Let us first calculate the mass of the cylinder. Let us consider a shell of the thickness dr such that we can consider the density is constant over the shell. Now the surface area of the shell is 2 dA = hr dr Hence the mass of the shell is 2 2 3 dm = r hdr = rr hdr = hr dr Hence the total mass of the cylinder is R 3 hR4 M = r dr = 4 0 Now the moment of inertia of the cylinder is R R 2 2 3 hR 6 hR 4 2 4 2 4 2 I = rdm = r r dr = 6 = 4 6R = M × R = 6R 6 0 0 2 Hence the moment of inertia will be MR . 4 6