Determining Moment of Inertia of a Sphere I'm having some troubles determining the moment of inertia of a sphere about it's central axis. My original question was to calculate it for a cylinder, which I've done, but I'd like to know how to find it for a sphere. Here is the problem solved for a cylinder: Problem: A uniform solid cylinder has radius R, mass M, and length L. Calculate its moment of inertia about its central axis (the z axis). Solution: I divided the cylinder into infinitesimally small layers because I knew that [tex]dV = (2\pi*r*dr)*L[/tex]. From here I calculated the integral [tex]I = \int \rho*r^2 dV = \int_{0}^R \rho*r^2*(2\pi*r*L)dr = 2*\pi*\rho*L*R^4 [/tex] I substituted [tex]\frac {M}{\pi*R^2*L}[/tex] (or [tex]\frac{M}{V}[/tex]) for [tex]\rho[/tex] into the equation to get [tex] I = \frac{1}{2}*\pi*(\frac {M}{\pi*R^2*L})*L*R^4 = \frac{1}{2}*M*R^2[/tex] I understand this, but when I tried to get it as a sphere I ended up getting the wrong answer. Could anyone please show me how to start the problem with a sphere?
Do it in spherical coordinates, being careful to take the sin(theta) for each little volume piece (since the spherical r value is from the origin, not from the axis of rotation). And be careful to use the correct value for dV in spherical coordinates.
How did you try to perform your integral for the sphere? Think of your choice of coordinate system--hint: spherical polar coordinates. damn: really should have refreshed quicker!
Couldn't I divide the sphere into a hemisphere and multiply it by 2? E.G. [tex]2*\pi*\int_{0}^R (\sqrt{R^2-r^2}*r^2)*r^2 dr[/tex] Where R is the radius of the sphere. So essentially I'm adding a bunch of [tex]\pi*r^2[/tex] (circles) to get a half of a sphere. Then multiplying the whole thing times two.
hmm, why dont you show a little more work so that it is easier to follow. one thing though, you might want to reconsider muliplying it by two--think about it.
I have no idea how to integrate that, so I don't have much work to show. The idea makes sence in my head, but I dont know how to follow through with it.