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I'm asked to find the center of mass to the semi-sphere to: $$x^2+y^2+z^2=1\quad z\geq0$$ I'm also told that the mass density is $1$.

I can make a parametrization of the semi-sphere as follows: $$r(\phi,\theta) = \sin(\phi)\cos(\theta)\hat{\textbf{i}}+\sin(\phi)\cos(\theta)\hat{\textbf{j}}+\cos(\phi)\hat{\textbf{k}}, \quad \phi \in \left[0, \frac{\pi}{2}\right],\theta \in \left[0, 2\pi\right]$$ But from here on i'm pretty clueless on how to setup double integrals for finding the coordinates. I don't even know if this is the best way to parameterize the sphere for this task.

Any help/pointers would be nice. Also please don't give me the solution as i would very much like to explore it for myself!:)

Thanks in advance

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  • $\begingroup$ What is your definition of center of mass? Is the definition you have not given in terms of integral expressions? $\endgroup$ Commented Feb 23, 2016 at 10:39

2 Answers 2

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A correct parameterization of the hemi-sphere $x^2 + y^2 + z^2 = 1$, $z \ge 0$, is:

$$\begin{cases} x = r \cos \theta \sin \varphi \\ y = r \sin \theta \sin \varphi \\ z = r \cos \varphi \end{cases} ; \begin{cases} 0 \le r \le 1 \\ 0 \le \varphi \le \pi/2 \\ 0 \le \theta \le 2 \pi \end{cases} $$

In general, given a solid $S$,

$$M = \int_S\rho \ d\sigma(P)$$

($P$ denotes a point on $S$)

And,

$$\vec{OG} = \frac1M \int_S \rho \ \vec{OP} \ d \sigma(P)$$

In this situation, we have $\rho = \rho(r,\theta,\varphi) = 1$ and $d \sigma(P) = r^2 \sin \varphi \ dr \ d\varphi \ d\theta$ (since $r^2\sin \varphi$ is the Jacobian of the transformation from cartesian to spherical coordinates; initially we would say $d \sigma(P) = dz \ dy \ dx$). So,

$$M = \int_0^{2\pi} \int_0^{\pi /2} \int_0^1 r^2 \sin \varphi \ dr \ d\varphi \ d\theta$$

$$\vec{OG} = \frac1M \int_0^{2\pi} \int_0^{\pi /2} \int_0^1 r^2 \sin \varphi \ \vec{OP} \ dr \ d\varphi \ d\theta$$

With $\vec{OP} = (x,y,z) = (r \cos \theta \sin \varphi, r \sin \theta \sin \varphi, r \cos \varphi)$

Now you just compute.

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If $\Sigma \subset \mathbb{R}^3$ is the domain which you want to find the center of mass $G=(G_x,G_y,G_z)$ then you have for an homogeneous repartition of the mass :

$G_x=\frac{1}{M}\underset{\Sigma}\iiint x dx dy dz$, $G_y=\frac{1}{M}\underset{\Sigma}\iiint y dx dy dz$,$G_z=\frac{1}{M}\underset{\Sigma}\iiint z dx dy dz$

Where $M$ is the mass, so $M=Vd$ where $d$ is the density and $V$ the volume. Using the spherical coordinates, try using thoose three integrals for your problem and $\Sigma$.

The idea behind is that finding the center of mass is the same as finding the expected value of a random variable following an uniform distribution which lives on the domain. (if the distribution is homogeneous).

Furthermore, you can also, by thinking about the symetries of your system, determine if it is relevant or not to calculate all thoose integrals.

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