How to Calculate Rotational Inertia for Different Objects?

[ahuebel]

I would like to further understand rotational inertia. I understand that for a point mass, I = MR^2 and for a continuous object it is basically the sum of all "little" MR^2 for each element of that object. I get a little fuzzy when actually solving for I for an object. For example, if we have a cone, the mass M of the cone is the density * volume or rho*(1/3)pi*r^2. So to find inertia we take the integral of the product of mass and R^2 but over what interval? My book says to take it over the volume but I am not 100% sure what that means. Would it be the triple integral dx dy dz (or more easily dr d(theta) dz)? If so, what would be the integrand?

TIA for any help

 

[arildno]

First, always be clear about where the axis of rotation is, relative to the position of the object.
If we let the z-axis denote the rotation axis, then the moment of inertia of the object is given by $$I=\int_{V}\rho(x^{2}+y^{2})dV$$
where x,y are orthogonal coordinates in a plane defined by the rotation axis.