(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the moment of inertia of a hollow sphere with mass m and radius R and uniform density

2. Relevant equations

Since the hollow sphere is an area, the density is mass divided by area, so:

[tex]I = \int r^2 dm = \frac{m}{A}\int r^2 dA[/tex]

3. The attempt at a solution

. The total area is 4pi r^2, so here is what I got

[tex]dA = 2\pi \sqrt{R^2-r^2}dr[/tex]

[tex]I = \frac{m}{4\pi R^2} \int_{-R}^{R} r^2(2\pi\sqrt{R^2-r^2})dr[/tex]

[tex]I = \frac{m}{R^2} \int_{0}^{R} r^2\sqrt{R^2-r^2}dr[/tex]

From here I made the substitution [tex]r = R\sin{\theta}[/tex] and got

[tex]I = mR^2 \int_{0}^{\frac{\pi}{2}} \sin^2\theta\cos^2\theta d\theta[/tex]

And that evaluated to pi/16, which brings me to my problem

the correct answer is supposed to be [tex]I = \frac{2mR^2}{5}[/tex]

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# Moment of inertia of a hollow sphere

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