Physics 263: Problem Set #7

1. Rotating cube. I found it convenient to put the origin at the center of the cube with the z-axis the rotation axis (this is not the only useful choice!). You know the cube is uniform, with total mass M and you know (presumably!) its volume given the length L of a side, so that means you know the (uniform) density. With straight edges at right angles everywhere, one of our coordinate systems is preferred. You can check your answer somewhat if you recall that the moment of inertia for a solid disk is M R^2/2. The "effective radius" here is somewhere between L/2 and L/sqrt(2) (the diagonal), so the moment of inertia should be between M L^2/8 and M L^2/4.
2. Solid cone. This should be straightforward after the example in class (use similar triangles to find the integration limits).
3. Charged hemisphere. The angle theta in the charge density expression is the polar angle and not the azimuthal angle (despite the opposite convention many of you know from Math 254) while r is the radial distance in spherical coordinates. (So the obvious choice of coordinates here is spherical.) Charge density here is charge per volume, so it is like the mass density; to find the total charge, sum up (i.e., integrate) the density times each little dV volume element.
4. Exponential sphere. The r here is again the radial distance in spherical coordinates. Note that the density is given in terms of rho0 but the answer should be in terms of M. So calculate M and use the result to replace rho0 in your expression for I by M.