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Homework Statement
Homework Equations
The Attempt at a Solution
Consider a very thin shell of width dr at a distance r from the center . The volume of this shell is ##4 \pi r^2 ## . Mass is ## m = 4 \pi r^2 \rho## .
P is the pressure at distance r . Gravitational acceleration at distance r < R is ##g' = \frac{GMr}{R^3}##
Doing a force balance on this thin shell .
##P(4 \pi r^2)  (P+dP) (4\pi r^2) = mg'##
##  dP (4\pi r^2) = (4 \pi r^2 \rho)\frac{GMr}{R^3}##
Integrating under proper limits ,
$$ P(r) = \frac{GM\rho}{2R} \left ( 1  \frac{r^2}{R^2} \right )$$ . V is the volume of the star .
Now , at the surface P(r = R) = 0 .
Is the question wrong ? Should the question be asking about the pressure "at the center" instead of "on the surface" ?
Thanks
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