Using Pappus's Guldinus theorem to find the mass centre Recently I have found an an interesting theorem which says that

surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length s of C and the distance d traveled by the geometric centroid of C: $A = sd$.

It is called the Pappus's centroid theorem.
I wonder if it is possible to find the mass centre using the theorem above. I can't find any example however. I would appreciate seeing one.
 A: I don't think that it is possible to find the center of mass using just the theorem above. If you know the surface area $A$ and the arc length $s$ of $C$, what you can do is to find the distance $d=A/s$ travelled by the geometric centroid of $C$. This distance corresponds to the length of the circumference described by the geometric centroid of $C$ while rotating curve $C$ around an axis. You can therefore find the radius $r=\frac{d}{2\pi}$, i.e. the distance of the curve's centroid from the solid's axis. The information you get is therefore that the curve's centroid lies somewhere in a straight line parallel to the solid's axis and having distance $r$ from it. But, using the theorem alone, you cannot determine the exact "vertical coordinate" of the centroid (assuming that the solid's axis is vertical). This vertical coordinate would correspond to the $z-$ coordinate of the solid's center of mass (which is, in turn, constrained to live on the solid's axis for symmetry reasons). 
In the following, I propose you an alternative way to find the solid's center of mass.  
At first, you have to check the following assumptions:


*

*Is your solid axial-symmetryc?

*Is the density constant in the solid?

*Do you know the expression of the curve $C = f(z)$?


If the answers are a triple "yes", I would do the following. First of all, you can recognize a priori that the center of mass will be located on the symmetry axis of your solid. Let's say, for example, that it is the vertical axis, $z$. So the only thing you've to compute is the $z-$ coordinate of the center of mass.
You can then compute the centroid of curve $C=f(z)$. The value you will obtain (let's call it $z^*$) is exactly the coordinate of the center of mass of the whole solid. 
Example: take a uniform cylinder, whose height is $h$ and whose radius is $r$. The solid is axially-symmetric and has constant density, so we can go on. Put the $z-$ axis on the cylinder's axis and set $z=0$ on the bottom face. The equation of the curve $C$ is, in this case, $f(z)=r$, for $z\in[0,h]$ (practically, you are describing an edge of the orthogonal projection of the cylinder). You can compute the $z-$coordinate of the centroid of $C$ in the following way:
$$
   z^* = \frac{\int_0^h zr\, \mathrm{d}z}{\int_0^h r\, \mathrm{d}z} = \frac{\frac{h^2 r}{2}}{hr} = \frac{h}{2}
$$
as expected also without computing the integral.
