Forece Between Two Hemispheres

  • Thread starter castusalbuscor
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In summary, the problem is to find the force necessary to pull two uniform hemispheres of mass M and radius a apart, which are placed in contact to form a complete sphere. After mulling over it for three days, the solution was found to be \frac{3M^{2}G}{4a^{2}}, with the help of another student who suggested approaching it through density. The final solution involved solving a complex integral, resulting in the formula 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 to the center of the sphere.
  • #1
castusalbuscor
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[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 that's useless to me if I can't get to it.

How would I go about to begin solving the problem?
 
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  • #2
Is the only force between them gravitational? Is that what we're overcoming here?
 
  • #3
Yes. the solution is [tex]\frac{3M^{2}G}{4a^{2}}[/tex]
 
  • #4
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:

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

Which gives [tex]F = \frac{3GM^{2}}{4a^{2}}[/tex]

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

1. What is the force between two hemispheres?

The force between two hemispheres is the gravitational force that exists between any two objects with mass. In this case, the two hemispheres are considered as point masses with their centers at the same distance from each other.

2. How is the force between two hemispheres calculated?

The force between two hemispheres can be calculated using the formula F = G(m1m2/r^2), where G is the gravitational constant, m1 and m2 are the masses of the two hemispheres, and r is the distance between their centers. This formula is derived from Newton's Law of Universal Gravitation.

3. What factors affect the force between two hemispheres?

The force between two hemispheres is affected by two main factors: the masses of the two hemispheres and the distance between their centers. As the masses increase, the force between them increases. However, as the distance between their centers increases, the force decreases.

4. Is the force between two hemispheres always attractive?

Yes, the force between two hemispheres is always attractive. This means that the two hemispheres will always be pulled towards each other due to their gravitational attraction.

5. How does the force between two hemispheres compare to other types of forces?

The force between two hemispheres is a non-contact force and is considered to be one of the weakest fundamental forces in nature. It is much weaker than other forces such as electromagnetic force and the strong and weak nuclear forces.

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