Half Shell Center of Mass

Calculus Level pending

A hollow spherical half-shell centered on the origin and defined only for $z \geq 0$ has an area-mass-density $\sigma = z$ . The radius of the shell is $1$ .

Find the $z$ location of the half-shell's center of mass.

Inspiration

The answer is 0.667.

1 solution

Otto Bretscher
Dec 17, 2018

Let $I_n=\int\int_S z^n\ dS=\int_0^{2\pi}\int_{0}^{\pi/2}\cos^n\phi\sin\phi \ d\phi\ d\theta=\frac{2\pi}{n+1}$ . Now $\overline{z}=\frac{I_2}{I_1}=\frac{2}{3}\approx \boxed{0.667}$ .

Half Shell Center of Mass

The answer is 0.667.

1 solution

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