Question

Gravitational potential at any point inside a spherical shall is uniform and is given by – (GM / R) where M is the mass of shell and R its radius. At the center solid sphere, potential is [– {(3GM) / (2R)}]

(1) There is a concentric hole of radius R in a solid sphere of radius 2R mass of the remaining portion is M what is the gravitational at center? 

(A) – [(3GM) / 7R] 

(B) – [(5GM) / 7R] 

(C) – [(7GM) / 14] 

(D) – [(9GM) / (14R)]

Answer 1

Correct option: (D) – [(9GM) / (14R)]

Explanation:

ρ = (M/V)

ρ = [M / {(4/3)πr³}]

= [M / {(4/3)π[(2R)³ – R²]}]

= [(3M) / (28πR³)]

V_whole = V_hole + V_remaining.

V_remaining = V_whole – V_hole 

= (3/2)[– (GM₁ / 2R) + (GM₂ / R)]

here   M₁ = (4/3)πρ(2R³) = (8/7)M

M₂ = ρ[(4/3)π]R³ = (1/7)M

V_remaining = (3/2)(G/R) × [– (M₁ / 2) + M₂]

= (3G / 2R)[{(– 8) / (2 × 7)}M + (M/7)]

= (3G / 2R) × – (3M / 7)

= – [(9Gm) / (14R)]