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I am studying for a calculus exam and am having trouble setting up this equation.

Let E be the solid sphere $x^2 + y^2 + z^2 \leq 1$ with a constant density. Find it's moment of inertia $I_z$ about the $z$-axis.

I am not looking for exact answers to the problem as much as just how to set up the problem for reference and studying. From what I know this should end up being an integral(s), but I am not sure of the start or end points for it. Thanks in advanced.

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2 Answers 2

The moment of inertia is given by

$$I_z = \frac12 \int dm\: \rho^2$$

where $dm = p\,dV$ is a mass element, $p$ is the constant density, $dV$ is a volume element, and $\rho$ is a distance from a point where the mass element is located to the axis about which the moment of inertia is to be computed. In this case, we have a sphere and $\rho^2 = 1-z^2 = 1-r^2 \cos^2{\theta}$. The integral is then

$$I_z = \frac12 2 \pi p \int_0^1 dr \, r^2 \: \int_0^{\pi} d\theta \, \sin{\theta} (1-r^2 \cos^2{\theta})$$

Evaluating this integral gives

$$I_z = 2 \pi p \int_0^1 dr \, r^2 \left ( 1 - \frac13 r^2 \right ) = 2 \pi p \frac{4}{15} = \frac25 M$$

where $M = 4 \pi p/3$ is the mass of the sphere.

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