The radii of two planets A and B are in the ratio 2 : 3. Their densities are 3ρ and 5ρ respectively. The ratio of their acceleration due to gravity is :

Solution

Physics sir · 3 steps

Write the mass as density times volume, and g turns out proportional to ρ R, not 1/R².

1. Mass from density. A planet of radius R and density ρ has mass 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. G and 4/3π cancel, so g_A/g_B = (ρ_A R_A)/(ρ_B R_B) = (3ρ × 2)/(5ρ × 3) = 6/15 = 2/5.
3. So. The ratio of accelerations due to gravity is 2 : 5, which is option (d).

The idea to keep. When densities are given instead of masses, use g = 4/3π Gρ R. Here g grows with radius, because a bigger planet carries more mass.

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