Prove that the horizontal force needed to pull two weightless, thin hemispheres apart has magnitude
[tex] F = \pi*r^2 * \Delta p [/tex]
where [tex] \Delta p [/tex]
is the pressure difference between the inside and outside of the spheres.
[tex] surface area = 4\pir^2
F = pA [/tex]
The Attempt at a Solution
I can see that the force needed to pull the hemispheres apart will be equal to the net force on one of the hemispheres along the X-axis created by the pressure (which is uniform).
I am not sure how to calculate the net force along the X axis on either hemisphere. If this were a ring in a plane and there were a simple force along the circumference towards the center of the ring, then I would integrate the x-components of the force elements dF along the circumference of a half ring from 0 to pi. But I am not sure what kind of integration I can do since I'm dealing with pressure and 3 dimensions. Thanks.
I apologize for the poor formatting, can someone tell me how to write this properly in Tex? You should be able to see my tex code by mousing over.