Two planets A and B have densities ^ ₁^ , ^ ₂^ and have radii ^ r ₁^ , ^ r ₂^ respectively. The ratio of acceleration due to gravity on A to that of B is

Step-by-step solution

The acceleration due to gravity at a planet's surface is g = GM R^2 , where G is the gravitational constant, M is the planetary mass, and R is the radius. The mass can be expressed in terms of density and volume as M = V . For a spherical planet, V = 4 3 R^3 , so M = 4 3 R^3 . Substituting into the gravity formula yields g = G 4 3 R^3 R^2 = 4 3 G R . For planet A: g_A = 4 3 G ₁ r₁ For planet B: g_B = 4 3 G ₂ r₂ The ratio simplifies to g_A g_B = ₁ r₁ ₂ r₂ . The acceleration ratio is r₁ ₁ : r₂ ₂ .