Find the moment of inertia of a thick cylindrical shell, moment of inertia integral.
00:00 In this physical integration example, we find the moment of inertia of a thick cylindrical shell of uniform density by setting up a moment of inertia integral. We are given the dimensions of the thick cylindrical shell: inner radius, a, outer radius, b, mass, M, length, L, and we are told to use nested thin cylindrical shells of mass dm to compute the total moment of inertia of the thick shell.
00:42 Expressing dm in terms of r and dr: our first task is to express the mass of a thin cylindrical shell of radius, r, in terms of r and dr (the variable for our moment of inertia integral because the thin cylindrical shells are varying with r). We start with a quick note on density: density is mass/volume so we can always find mass by taking mass=density*volume. Now we cut and unroll our thin cylindrical shell of radius r and find the volume of the shell as 2pi*r*L*dr. Multiplying by the density, rho, we find the mass increment given by the thin shell: 2pi*rho*L*r*dr.
02:35 Moment of inertia contribution of dm: now we can write down the moment of inertia contribution for the thin cylindrical shell of mass dm. We appeal to our previous result for the moment of inertia of a thin cylindrical shell: I=mr^2. The derivation of this fact can be found here: • Derivation: moment of inertia for a ... . We simply plug in the expression for dm and simplify the result to get dI=2pi*rho*L*r^3*dr.
03:03 Integrate to get the total moment of inertia: with an expression for dI entirely in terms of the integration variable, r, we can now integrate as r goes from a to b to sum the moment of inertia contributions of all cylindrical shells from the inner radius, a, to the outer radius, b. We get a final answer for moment of inertia, but it still has a rho (density) in it. We need to eliminate rho by taking the mass of the thick cylindrical shell divided by the volume of the thick cylindrical shell. The volume is given by the area of the base (pi*b^2-pi*a^2) multiplied by the length L. After substituting for density in the moment of inertia and using a factoring trick, we arrive at the moment of inertia for a thick shell formula: 1/2M(b^2+a^2).
