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A uniform sphere has a mass M and radius...

A uniform sphere has a mass M and radius R. Find the pressure p inside the sphere, caused by gravitational compression as a function of the distance r from its centre.

Text Solution

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Consider a layer of thickness `dr` at a distance `r` from the centre of the sphere.
Now force due to the layer `dF=4pir^(2)dP`
`impliesdPrpir^(2)=(G((4)/(3)pir^(3)rho)(4pir^(2)drrho))/(r^(2))`
(Where `rho` is the mean density of sphere)
or, `dP=G(4)/(3)pirho^(2)rdr` `:. p=G int_(r )^(R )(4)/(3)pirho^(2)rdr`
or `P=G(2pi)/(3)rho^(2)(R^(2)-r^(2))`
`=(3)/(8)F({1-r^(2)//R^(2)}M^(2))/(piR^(4)) [ :. rho=M//(4//3)piR^(3)]`
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