Two planets A and B of radii R and 1.5 R have densities ρ and ρ / 2 respectively. The ratio of acceleration due to gravity at the surface of B to A is:

Solution

Physics sir · 3 steps

Write the mass as density times volume, and g on the surface comes out proportional to ρR.

1. Mass from density. For a uniform sphere, M = ρ × 4/3π R³. Put this into g = GM/R².

   g = G/R² × 4/3π R³ ρ = 4/3π G ρ R

2. Take the ratio. Everything except ρ R is the same for both planets, so it cancels: g_B/g_A = (ρ/2)(1.5R)/(ρ R) = 0.75/1 = 3/4.
3. So. The ratio of g at the surface of B to that of A is 3 : 4, option (d).

The idea to keep. When planets are compared by density, use g = 4/3π Gρ R, so g ∝ ρ R. Use g = GM/R² only when the masses are given.

Answer from the NTA final answer key. Solution by PrepWiser.