Simple derivation: sphere contracting under gravity

In summary, the conversation is about a reader asking for help in deriving a formula mentioned in Roger Penrose's book "The Road to Reality". The formula in question is "The rate of volume reduction is 4πGM", which the reader is trying to understand. They discuss the volume of a sphere and the acceleration due to Newton's law of gravity, and try to find a way to combine these to get the desired formula. Ultimately, they are able to figure out that multiplying 4πr2 by GM/r2 gives the correct answer, but they are unsure if this is a valid method.
  • #1
HowardTheDuck
33
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Hi Guys, I'm reading Roger Penrose's book "The Road to Reality" at the moment and I wonder if you could help me out with a pretty simple derivation which he doesn't describe in complete detail.
On page 399 he considers a sphere of mass contracting under gravity, and says "The rate of volume reduction is 4πGM". Can you help me derive this, please? It shouldn't be too difficult, it's just beyond me!

OK, what do we know. The volume of a sphere is 4/3πr3. So dV/dr is 4πr2.
And the acceleration (d2r/dt2) due to Newton's law of gravity would be GM/r2. Can we combine these via the chain rule or similar to get the rate (acceleration?) of volume reduction equal to 4πGM.

Thanks a lot.
 
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  • #2
I can see that if you simply multiply the 4πr2 by the GM/r2 then you get the right answer. But I don't see how that's a valid thing to do.
I don't see how it's valid to multiply dV/dr by d2r/dt2 (by the chain rule?) to get the right answer.
Any help appreciated.
 

1. How does gravity affect the contraction of a sphere?

Gravity is the force that pulls objects towards each other. In the case of a sphere contracting under gravity, the force of gravity acts towards the center of the sphere, causing it to shrink in size.

2. What factors determine the rate of contraction of a sphere under gravity?

The rate of contraction of a sphere under gravity depends on the mass of the sphere, the distance between the center of the sphere and the object it is contracting towards, and the strength of the gravitational force between the two objects.

3. How does the shape of the sphere affect its contraction under gravity?

The shape of the sphere does not have a significant impact on its contraction under gravity. As long as the sphere is symmetrical and the force of gravity is acting towards its center, it will contract at the same rate regardless of its shape.

4. Can a sphere ever stop contracting under gravity?

No, a sphere will continue to contract under gravity until it reaches a point where the force of gravity is balanced by another force, such as the internal pressure of the sphere. At this point, the sphere will reach a state of equilibrium and stop contracting.

5. How is the contraction of a sphere under gravity related to the formation of celestial bodies?

The process of a sphere contracting under gravity is similar to the process of accretion, where smaller particles come together to form larger objects like planets and stars. The force of gravity acts to pull these particles towards each other, causing them to contract and eventually form a spherical shape.

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