Consider a halved solid torus (half a donut). The radius of the torus are $R_1$ and $R_2$. I need to find its center of mass. The hint they give is that the center of mass of a homogeneous solid object $Omega subset Bbb R^3$ is calculated as$$overlinex=int_Omegaoverlinerdoverliner.$$

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I really don"t understand this formula, I don"t know what $overliner$ means and what is $Omega$ in this case. I"d appreciate that someone explains what this formula means and how to apply it in this problem. Thanks in advance.




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asked Oct 19 "13 at 0:01
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I found it easiest to use cylindrical coordinates to set up the integrals needed for the center of mass. Before we do so, however, I set my coordinate system up as follows. I have positive $x$ coming out of the screen, positive $y$ going to the right, and positive $z$ up. In cylindrical coordinates $(r,phi,z)$:

$$x = r cosphi$$$$y=r sinphi$$

where we have the limits defining the region $Omega$:

$$phi in left < -fracpi2,fracpi2 ight>$$$$z in <-R_2,R_2>$$$$r in left < R_1 - sqrtR_2^2-z^2, R_1 + sqrtR_2^2-z^2 ight>$$

Also, for an object of constant mass density, the expression for the $x$ component of the center of mass is

$$arx = fracdisplaystyle int_Omega d^3 vecx , xdisplaystyle int_Omega d^3 vecx$$

(Note that, by symmetry, we have $ary=0$ and $arz=0$.) Let"s first evaluate the denominator:

$$eginalign int_Omega d^3 vecx &= int_-pi/2^pi/2 dphi , int_-R_2^R_2 dz , int_R_1 - sqrtR_2^2-z^2^R_1 + sqrtR_2^2-z^2 dr , r \ &= fracpi2int_-R_2^R_2 dz left \ &= 4 pi R_1 int_0^R_2 dz , sqrtR_2^2-z^2\ &= pi^2 R_1 R_2^2 endalign$$

Now we evaluate the center of mass:

$$eginalignarx &= frac1pi^2 R_1 R_2^2 int_-pi/2^pi/2 dphi , int_-R_2^R_2 dz , int_R_1 - sqrtR_2^2-z^2^R_1 + sqrtR_2^2-z^2 dr , r^2 cosphi \ &= frac43 pi^2 R_1 R_2^2 int_0^R_2 dz , left \ &= frac83 pi^2 R_1 R_2^2 int_0^R_2 dz ,left <3 R_1^2 sqrtR_2^2-z^2 + left (R_2^2-z^2 ight )^3/2 ight > \ &= frac83 pi^2 R_1 R_2 left ( frac3 pi4 R_1^2 R_2 + frac3 pi16 R_2^3 ight )endalign$$

Simplifying, I get

$$arx = frac4 R_1^2+R_2^22 pi R_1$$

ADDENDUM

As a quick note, in the limits as $R_2 o 0$, we find that the center of mass becomes