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I'm trying to set the ratio between the variation of the diameter cutting in the region lower a sphere with respect to distance from the center of mass with the center of the radius of the sphere.

For example, if the diameter is zero the center of mass is zero. If the diameter lower is equal to the diameter of the sphere, the Center of mass is $1.875$.

enter image description here

I have this equation, but this is relative to $H$.

enter image description here

Using trigonometry could find this reason. But was thinking to have a answer in terms of integration or derivation.

I wanted to understand how was obtained the above equation. I believe that has been by integration.

Something like this: Equation

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  • $\begingroup$ Would you please write more details? $\endgroup$ – Arman Malekzadeh May 28 '17 at 14:38
  • $\begingroup$ Please show your integration steps. $\endgroup$ – Narasimham May 28 '17 at 14:46
  • $\begingroup$ An easy way to derive it is to find the center of mass of a circular segment and subtract. $\endgroup$ – amd May 28 '17 at 18:01
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To try to solve I applied a simple trigonometry:

enter image description here

The only equation I know is this:

$y=\frac{3 (2 r-H)^2}{4 (3 r-H)}$

Instead of using $H$ I intend to use $Ø$. With this I made a simple replacement:

$h=\sqrt{r^2-\left(\frac{\text{Ø}}{2}\right)^2}$

$H=h+r$

Therefore:

$y=\frac{3 \left(r-\sqrt{r^2-\frac{\text{Ø}^2}{4}}\right)^2}{4 \left(2 r-\sqrt{r^2-\frac{\text{Ø}^2}{4}}\right)}$

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  • $\begingroup$ I have presented this answer to show what I want to achieve. I want the relationship between the bottom diameter and the center of mass, but I would like to get that result another way. Something that was not based on the equation with $H$. $\endgroup$ – LCarvalho May 29 '17 at 15:12

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