Using rings to find the moment of inertia of a hollow sphere (physical integration).

00:00 We compute the moment of intertia of a thin spherical shell by slicing the shell into thin rings. The strategy is to find the mass and radius of each infinitesimal ring, then use the formula for the moment of inertia of a ring derived previously here:    • Derivation:  moment of inertia for a ...  

00:38 A note on area density: we introduce the idea of area density for a surface (the mass per unit area, or mass divided by area). The area density for a sphere is M/4piR^2 for the sphere, and we can also say that mass is area density multiplied by area. This is also true for the differential area of the thin ring, so we can get the infinitesimal mass of the ring by multiplying the area density sigma by the area dA.

01:34 Deriving the area of the thin ring as a function of theta: we label the dimensions of the thin ring, starting with the radius of the sphere connecting the center of the sphere to the edge of the ring. We also label the angular position of the ring by labeling an angle theta with respect to the horizonal. We find the thickness of the ring as an infinitesimal increment of arc ds=Rd(theta), and the radius of the ring is given by Rcos(theta). Next, we cut and unroll the ring to get a thin rectangle, and we compute the infinitesimal area of this rectangle. Finally, we multiply the area by area density to get the mass of the thin ring, dm.

03:40 Moment of inertia contribution for a single thin ring: now that we have the mass of the thin ring, we use the standard formula for the moment of inertia of a ring: I=mr^2 and sub in our expressions for dm and r. This results in our final expression for the moment of inertia of the thin ring. We note that the integration variable is theta, and the bounds on theta are -pi/2 to pi/2 to cover all the rings from the bottom of the sphere to the top.

04:59 Physical integration: adding up the moment of inertia contributions to compute the moment of inertia of a thin spherical shell about its diameter. The total moment of inertia is given by the integral of the moment inertia contributions of the thin rings. This results in an integral of cosine cubed on an interval symmetric about the origin. We begin by using the parity of the cosine function to split the integration interval, then we use the standard substitution 1-sin^2(theta) to replace two factors of the cosine function. Using the chain rule backwards, we evaluate the antiderivatives and arrive at an expression for the moment of inertia in terms of the area density of the spherical surface. When we replace the area density with M/4piR^2, we arrive at the standard formula for the moment of inertia of a hollow ball 2/3MR^2 by using rings to find the moment of inertia of a hollow sphere.