Two planets A𝐴 and B𝐵 of radii R𝑅 and 1.5R1.5𝑅 have densities ρ𝜌 and ρ/2𝜌/2 respectively. The ratio of acceleration due to gravity at the surface of B𝐵 to A is

The acceleration due to gravity on the surface of a planet is given by the formula:

g = GM/R^2

where G is the gravitational constant, M is the mass of the planet, and R is the radius of the planet.

The mass of a planet can be found by the formula:

M = ρV

where ρ is the density of the planet and V is the volume of the planet.

The volume of a planet can be found by the formula:

V = 4/3πR^3

So, the mass of planet A is:

M_A = ρ * 4/3πR^3

and the mass of planet B is:

M_B = (ρ/2) * 4/3π(1.5R)^3

The acceleration due to gravity on the surface of planet A is:

g_A = G * M_A / R^2 = G * ρ * 4/3πR^3 / R^2 = 4/3 GρπR

The acceleration due to gravity on the surface of planet B is:

g_B = G * M_B / (1.5R)^2 = G * (ρ/2) * 4/3π(1.5R)^3 / (1.5R)^2 = 2/3 GρπR

So, the ratio of the acceleration due to gravity at the surface of planet B to that of planet A is:

g_B / g_A = (2/3 GρπR) / (4/3 GρπR) = 1/2

Therefore, the acceleration due to gravity at the surface of planet B is half that of planet A.