The mass density of a planet of radius R varies with the distance r from its centre as ρ ( r ) = ρ 0 1 - r 2 R 2 Then the gravitational field is maximum at:

Step-by-step solution

dm = ρ × 4 πx 2 dx = ρ 0 1 - x 2 r 2 × 4 πx 2 dx m = 4 πρ ∫ 0 r x 2 - x 4 R 2   dx   s m = 4 πρ 0 r 3 3 - r 5 5 R 2 E = Gm r 2 = G r 2 × 4 πρ 0 r 3 3 - r 5 5 R 2 E = 4 πGρ 0 r 3 - r 3 5 R 2 E is maximum when dE dt = 0     ⇒     dE dr = 4 πGρ 0 1 3 - 3 r 2 5 R 2 = 0 ⇒       r = 5 3 R E max = 4 πGρ 0 × 5 R 3 1 3 - 1 5 × 5 9 E max = 8 5 27 πGρ