# Calculate the position of the center of mass

1. Mar 2, 2015

### Robin04

1. The problem statement, all variables and given/known data
I would like to calculate the position of the center of mass of the body on the figure.
http://kepfeltoltes.hu/150302/161353224K_pkiv_g_s_www.kepfeltoltes.hu_.jpg

2. Relevant equations

3. The attempt at a solution
I read a way of solution: Theorem of Pappus
It states that if we rotate our figure around an axis, the volume of the body we got will be equal to the product of area of the 2D figure and the distance travelled by the center of mass.

The axis I chose is at the right bottom tip of the figure, and it is vertical.
I determined the area of this figure: area of rectangle ((side = 2R) - area of circle (radius = R))/4 = (R^2*(4-π))/4
The distance travelled by the center of mass is 2πx.
The volume of the body: (volume of cube (side = 2R) -volume of shere (radius = R))/2 = (R^3(9-(4/3)π))/2
Now comes the theorem:
(R^2*(4-π))/4 * 2πx = (R^3(9-(4/3)π))/2

I got for x = (R(9-(4/3))/((4-π)π)

But if I try it with an example where R = 1 m I got for x = 1,78 m which is outside and completely non-intuitive. Can someone help me?

2. Mar 2, 2015

### DEvens

You got the volume wrong. It is not a cube minus a sphere. Not even a portion of a cube minus a sphere. Yes, that's a circular arc. You get part of a sphere when you rotate a cicular arc if you are rotating around the centre of the circle. But you don't get a cube when you rotate a square around an axis. Maybe you can get the answer as a cylinder minus a sphere.

3. Mar 2, 2015

### Robin04

Thank you ver much. :)