# Use triple integral to find center of mass

## Homework Statement

Find the centre of mass of a uniform hemispherical shell of inner radius a and outer radius b.

## Homework Equations

##r_{CoM} = \sum \frac{m\vec{r}}{m}##

## The Attempt at a Solution

Using ##x(r,\theta,\phi)## for coordinates,
$$x_{CoM}=\frac{\int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} \vec{x}\rho r^2\sin{\theta}drd\theta d\phi}{\int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} \rho r^2\sin{\theta}drd\theta d\phi}$$
My vector calculus is rusty, how do I handle the ##\vec{x}## in this integral?

andrewkirk
Homework Helper
Gold Member
hvIt may help to first note that the LHS of the equation should be ##\vec x_{CoM}##. That is, it is a 3D vector that gives the location of the CoM.
Given that, treat the RHS as three separate integrals, one for each of the cartesian coordinates of the CoM: ##x,y## and ##z##.

##x_{CoM} = \int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} x(r,\theta,\phi) \rho r^2\sin{\theta}\,dr\,d\theta\, d\phi\ /\ D##
##y_{CoM} = \int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} y(r,\theta,\phi) \rho r^2\sin{\theta}\,dr\,d\theta\, d\phi\ /\ D##
##z_{CoM} = \int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} z(r,\theta,\phi) \rho r^2\sin{\theta}\,dr\,d\theta\, d\phi\ /\ D##
where ##D## is the denominator in the formula in the OP.
Then use the formulas for converting from Cartesian to spherical coords to replace the ##x(r,\theta,\phi)## in the integrand by a function of ##r,\theta,\phi##. THen do the same for the integrals for ##y## and ##z##.

BearY
hvIt may help to first note that the LHS of the equation should be ##\vec x_{CoM}##. That is, it is a 3D vector that gives the location of the CoM.
Given that, treat the RHS as three separate integrals, one for each of the cartesian coordinates of the CoM: ##x,y## and ##z##.

##x_{CoM} = \int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} x(r,\theta,\phi) \rho r^2\sin{\theta}\,dr\,d\theta\, d\phi\ /\ D##
##y_{CoM} = \int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} y(r,\theta,\phi) \rho r^2\sin{\theta}\,dr\,d\theta\, d\phi\ /\ D##
##z_{CoM} = \int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} z(r,\theta,\phi) \rho r^2\sin{\theta}\,dr\,d\theta\, d\phi\ /\ D##
where ##D## is the denominator in the formula in the OP.
Then use the formulas for converting from Cartesian to spherical coords to replace the ##x(r,\theta,\phi)## in the integrand by a function of ##r,\theta,\phi##. THen do the same for the integrals for ##y## and ##z##.
I am confused about what Cartesian coordinates do here. I thought I am just looking for a point in the spherical coordinate?

andrewkirk
Homework Helper
Gold Member
I am confused about what Cartesian coordinates do here. I thought I am just looking for a point in the spherical coordinate?
Quite right. Sorry, I was distracted by the use of the character 'x' in the vector ##\vec x##.
So we can do it more simply as:

##r_{CoM} = \int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} r \,\rho r^2\sin{\theta}\,dr\,d\theta\, d\phi\ /\ D##
##\theta_{CoM} = \int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} \theta\, \rho r^2\sin{\theta}\,dr\,d\theta\, d\phi\ /\ D##
##\phi_{CoM} = \int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} \phi\, \rho r^2\sin{\theta}\,dr\,d\theta\, d\phi\ /\ D##

BearY
Quite right. Sorry, I was distracted by the use of the character 'x' in the vector ##\vec x##.
So we can do it more simply as:

##r_{CoM} = \int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} r \,\rho r^2\sin{\theta}\,dr\,d\theta\, d\phi\ /\ D##
##\theta_{CoM} = \int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} \theta\, \rho r^2\sin{\theta}\,dr\,d\theta\, d\phi\ /\ D##
##\phi_{CoM} = \int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} \phi\, \rho r^2\sin{\theta}\,dr\,d\theta\, d\phi\ /\ D##
When I was studying vector calculus we only did parameterizing dimention x, y, and z with t and then integrate w.r.t dt only.
So without parameterization, we still integrate each dimension separately? That makes sense. I suppose I will have to redo vector calculus to know for sure.

vela
Staff Emeritus
Homework Helper
Quite right. Sorry, I was distracted by the use of the character 'x' in the vector ##\vec x##.
So we can do it more simply as:

##r_{CoM} = \int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} r \,\rho r^2\sin{\theta}\,dr\,d\theta\, d\phi\ /\ D##
##\theta_{CoM} = \int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} \theta\, \rho r^2\sin{\theta}\,dr\,d\theta\, d\phi\ /\ D##
##\phi_{CoM} = \int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{a}^{b} \phi\, \rho r^2\sin{\theta}\,dr\,d\theta\, d\phi\ /\ D##
I don't think this works because generally ##\langle \cos\theta \rangle \ne \cos\langle \theta \rangle##. Also, for a hemisphere of radius ##R##, the center of mass is on the z-axis at ##z=\frac {3}{16}{R}##, but the integral above with ##a=0## and ##b=R## gives ##r_{CoM} = \frac 38 R##.

BearY
LCKurtz
Homework Helper
Gold Member
@BearY: If that was a typical homework problem I would expect the student to consider the upper hemisphere at the origin, notice immediately that ##\bar x = \bar y = 0## from symmetry, and calculate$$\bar z = \frac {\iiint_V z~dV}{\iiint_V 1~dV} = \frac {\int_0^{2\pi}\int_0^{\frac \pi 2} \int_a^b \rho\cos\phi \rho^2\sin\phi~d\rho d\phi d\theta} {\int_0^{2\pi}\int_0^{\frac \pi 2} \int_a^b 1 \rho^2\sin\phi~d\rho d\phi d\theta}$$
Notice the integral is conveniently worked in spherical coordinates, but you are calculating the center of mass ##(\bar x, \bar y, \bar z)##. And it generally is not true, for example, that ##\bar z = \bar \rho \cos\bar\phi## just because ##z =\rho \cos\phi##. You must set up the rectangular integral then change coordinates.

BearY
Ray Vickson