Moment of Inertia with Variable Density Function

[Piamedes]

Homework Statement

There are two parts to question, the first asks for you to find the moment of inertia I for a thin disk of uniform density, a relatively trivial problem.

My problem centers around that second part, "Repeat the case where the density increases linearly with r, starting at 0 at the center, but the object has the same mass as the original disk."

Homework Equations

$$I = \int_{object} \rho (r,\theta) r^3 dr d\theta$$

The Attempt at a Solution

Assuming that the density function is p=kr, where k is some constant I'll work out later, then the moment of inertia would be:

$$I = \int_{0}^{2 \pi} \int_{0}^{R} k r^4 dr d\theta$$

$$I = k \int_{0}^{2 \pi} d\theta \int_{0}^{R} r^4 dr$$

$$I = 2 \pi \frac{r^{5}}{5} ]_{0}^{R}$$

$$I = \frac{2k\pi R^{5}}{5}$$

With this in mind I now would need to find k. I know that it must have units of kg/m^3 in order to make the moment of inertia have the proper units. My guess on how to do this is to integrate to find the total mass, which I know to be M, solve for k in terms of M and than back substitute:

$$M = \int dm$$

$$M = \int \rho dA$$

$$M = \int_{0}^{2 \pi} \int_{0}^{R} kr * rdrd\theta$$

$$M = k \int_{0}^{2 \pi} d\theta \int_{0}^{R} r^2 dr$$

$$M = 2\pi k \frac{r^3}{3} ]_{0}^{R}$$

$$M = \frac{2k\pi R^3}{3}$$

Solving for K:

$$k = \frac{3M}{2\pi R^{3}}$$

Now plugging that back into the equation for I,

$$I = \frac{2\pi R^{5}}{5} k$$

$$I = \frac{2\pi R^{5}}{5} \frac{3M}{2\pi R^{3}}$$

$$I = \frac{3MR^{2}}{5}$$

Is this the proper way to solve a moment of inertia problem of variable density?

Thanks for any and all help.

 

[Dick]

It looks just fine to me.

 

[Piamedes]

Thanks