So I am given the figure shown below and told to find the moment of inertia if we have that the mass of the shaded region is $M$. I think I have to find the total mass without the hole and the mass taken away for the hole and then subtract the moment of inertia of $r$ from the moment of inertia of $R$ using parallel axis theorem. I have no idea how to find the total mass however and am stuck. Any help is greatly appreciated! Thanks.
Best Answer
HINT: Density is an intrinsic property of a body.It is independent of how much of the material is present and is independent of the form of the material, e.g., one large piece or a collection of small particles.
Here the mass is based on area of the disc and not the volume.
Let the density of the disc be equal to $d$.
Then, Mass of total disc initially is $$ M_T=\pi R^2 d\\ $$
Mass Removed is, $$ m=\pi r^2 d\\ $$ We also have, $$ M=\pi (R^2-r^2)d $$ Therefore, $$ M_T=\frac{M\pi R^2}{\pi(R^2-r^2)} $$