The radii of two planets are respectively R1 and R2 and their densities are respectively ρ1 and ρ2 . The ratio of the accelerations due to gravity at their surfaces is

The correct option is g1:g2 = R₁ρ1: /R₂ρ2 (which simplifies to g₁:g₂ = R₁ρ₁ : R₂ρ₂).

To find the ratio of the accelerations due to gravity at the surfaces of the two planets, we begin with the standard formula for acceleration due to gravity on a planet's surface:

$g=\frac{GM}{R^2}$

where:
- $G$ is the universal gravitational constant,
- $M$ is the mass of the planet, and
- $R$ is the radius of the planet.

Assuming the planets are spherical, the mass ($M$) can be expressed in terms of volume and average density ($\rho$):

$M=\text{Volume}\times \text{Density}=\frac{4}{3}\pi R^3\rho$

Now, substitute this expression for mass ($M$) back into the formula for acceleration due to gravity ($g$):

$g=\frac{G\times \frac{4}{3}\pi R^3\rho }{R^2}$

Simplifying the powers of $R$, we obtain:

$g=\frac{4}{3}\pi GR\rho$

Since $G$ and $\frac{4}{3}\pi$ are constants, the acceleration due to gravity is directly proportional to the product of the planet's radius and its density:

$g\propto R\rho$

Therefore, the ratio of the accelerations due to gravity for the two planets is:

$\frac{g_1}{g_2}=\frac{R_1{\rho }_1}{R_2{\rho }_2}$

Expressing this as a ratio gives:

$g_1:g_2=R_1{\rho }_1:R_2{\rho }_2$