Cutting a hollow sphere into 2 piece but not in half

1. Oct 16, 2011

athrun200

1. The problem statement, all variables and given/known data
A uniform sphere of radius R is cut into two portions. The smaller portion S has height h measured from its flat surface, where h<R.

(a) Locate the center of mass of S from its flat surface

(b) Use (a) the locate the centre of mass of a hollow H which has the same shape and same size as S. The hollow has only a thin curved surface and without any flat surface.
(c) Obtain the result of (b) by direct integration.
2. Relevant equations
This question use 2 tricks to find the center of mass of a hollow.
1. Use substraction to substract a big object by a smaller one and then take limit. (I hope you understand what I mean.)
2. Use direction integration.

3. The attempt at a solution
My answer in part b doesn't agree with part c, can anyone help me to check what happen?

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2. Oct 16, 2011

vela

Staff Emeritus
I found a different expression for the mass of the cap:
$$m=\frac{\rho\pi h^2(3R-h)}{3}$$

3. Oct 16, 2011

athrun200

In fact I have same expression as you
$\frac{h^{3}}{3}$-$h^{2}R$

But when I copy it form messy worksheet to a much neat writting, I try to simplfy the expression in my mind, unforunately, I fail:tongue2:

So my expression for the mass is also correct.

Last edited: Oct 17, 2011
4. Oct 16, 2011

vela

Staff Emeritus
Did you get it to work out?

5. Oct 16, 2011

athrun200

Yes, of course.
I do this question with 2 other classmates, we check a hundred time for our answer. I am sure we get it correct.

6. Oct 22, 2011

tim94131358

so may i ask for b) and c)

the ans is..h/2 or h(4R-4) / 3(3R-h)

cuz my ans is not same as yours...

btw where the ans for b and c are not the same

7. Oct 22, 2011

vela

Staff Emeritus
The answers you get for b and c should both be h/2.

8. Oct 22, 2011

athrun200

what do you get?