Question

n Given that the moment of inertia about a diameter of a uniform hollow thin spherical shell of mass m and radius r is jmr2. Show that the moment of inertia about a diameter of a non-uniform sphere of radius R with the volume mass density distribution given by por) (1-), where r denotes the radial distance from the center, is 1 MR2. (Hint: Imagine the sphere is made of infinitely many layers of 15M 8TtR3 very thin concentric spherical shells.)

Answer 1

Consider a shell of radius 'r' and thickness 'dr' as shown in the figure Mass of shell, dm = rho times 4 pi r^2 dr dm = 15M/8 pi R^3 (1 - r^2/R^2) times 4 pi r^2 dr Therefore Moment of Inertia of thin shell dI = 2/3 dm r^2 dI = 5M/R^3 (r^4 - r^6/R^2) dr Integrating: I = 5M/R^3 integral^R_0 (r^4 - r^6/R^2)dr = 5M/R^3 [r^5/5 - r^7/7R^2]^R_0 I = 5M/R^3 times [R^5/5 - R^5/7] = 2/7 MR^2 Ans