# Forece Between Two Hemispheres

#### castusalbuscor

[SOLVED] Forece Between Two Hemispheres

I am trying to study for my classical mechanics final, and the prof hinted that this problem would be on the final.

Two uniform rigid hemispheres eachof mass M and radius a are placed in contact with each other so as to form a complete sphere. Find the forces necessary to pull the hemishperes apart

I've been mulling over it for three days now trying to figure out how to solve it.
I tried used spherical cooridnated to solve it but I am not sure where to begin.

I possibly could find the force due to a sphere of Mass 2M, but that does not help me in finding the force required to separate the hemispheres.

The back of the book gives the soloution, but thats useless to me if I can't get to it.

How would I go about to begin solving the problem?

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#### Locrian

Is the only force between them gravitational? Is that what we're overcoming here?

#### castusalbuscor

Yes. the solution is $$\frac{3M^{2}G}{4a^{2}}$$

#### castusalbuscor

So today I was bouncing some ideas off one of the smart kids in my class, and was asking for some direction.
The way he suggested I approach it was through density.

The big arse integral I had to integrate which give the required answer was:

$$F = \int\frac{4GM^{2}r^{3}sin\phi cos\phi d\phi d\theta dr}{4/3 \pi r^{6}}$$

Which gives $$F = \frac{3GM^{2}}{4a^{2}}$$

Where:
a is the radius of the hemisphere
M is the mass of the hemisphere
and r is the distance from the hemisphere

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