A spherical body of radius R consists of a fluid of constant density and is in equilibrium under its own gravity. If P(r) is the pressure at r(rltR), then the correct option(s) is (are)

Options

A. (a) `P(r=0)=0`
B. (b) `(P(r=3R//4))/(P(r=2R//3))=63/80`
C. (c) `(P(r=3R//5))/(P(r=2R//5))=16/21`
D. (d) `(P(r=R//2))/(P(r=R//3))=20/27`

Answer

A, B

Explanation

Let us consider an elemental mass dm shown in the shaded portion.
Here
`P4pir^2-(P+dP)4pir^2`
`=(GMr)/(R^3)rho(4pir^2)dr`
`:. -int_0^Pdp=(GMrho)/(R^3)int_R^rrdr`
`: P=(GMrho)/(2R^3)[R^2-r^2]`
`:. (P(r=3R//4))/(P(r=2R//3))=[(R^2-(9R^2)/(16))/(R^2-(4R^2)/(9))]=((7R^2)/(16))/((5R^2)/(9))=63/80`
and `(P(r=3R//5))/(P(r=2R//5))=[(R^2-(9R^2)/(25))/(R^2-(4R^2)/(25))]=16/21`
B and C are correct options.