Find the moment of inertia of a solid sphere by integrating thin spherical shells.

00:00 Given the moment of inertia of a thin spherical shell, we find the moment of inertia of a solid sphere by representing the sphere with nested spherical shells then integrating thin spherical shells to derive the moment of inertia of a sphere. Contrary to the previous approaches to this moment of inertia integral (moment of inertia by integrating over disks or cylindrical shells), our spherical shells match the symmetry of the problem perfectly, so we expect a much simpler moment of inertia integral to result from using spherical shells to compute the moment of inertia of the solid sphere.

00:58 Labeling a spherical shell and expressing the mass of the spherical shell in terms of density and volume. We label a representative spherical shell with mass dm, radius r and thickness dr, then we express the mass of the shell as density times volume: rho*dV, since the volume is infinitesimal.

01:46 How to compute the volume of a thin spherical shell: the physics way vs. the math way to find the volume of a spherical shell. The volume of a thin spherical shell can be thought of as surface area times thickness or 4piR^2*dr. However, this approach seems a little flimsy, so we justify the result more rigorously by taking the volume of the outer sphere minus the volume of the inner sphere to find the volume of the thin spherical shell. We argue that higher powers of dr are negligible since dr is an infinitesimal quantity, and we arrive at the same expression for dV.

04:01 Final setup of the moment of inertia integral for the solid ball. Now that we have an expression for dV, we multiply by the density rho to obtain an expression for the mass of the shell, dm. We plug dm into the given formula for the moment of inertia of a thin spherical shell and obtain an expression for the moment of inertia of a single thin shell dI. Finally, we use an integral to add up all the dI contributions.

05:18 Calulation of moment of inertia integral: now that the moment of inertia integral is expressed entirely in terms of the thin shell radius r, we can find the moment of inertia of a solid sphere by integrating over thin spherical shells. The integral is trivial, requiring only the power rule, and we arrive at an expression for the moment of inertia of a solid sphere. After substituting M/4/3*pi*R^3 for the density of the sphere, we arrive at the familiar formula for moment of inertia of a solid ball: 2/5*MR^2.