The diameters of two planets are in the ratio 4 : 1 and their mean densities in the ratio 1: 2. The acceleration due to gravity on the planets will be in ratio

To find the ratio of the acceleration due to gravity on the two planets, we can use the formula for acceleration due to gravity ($g$) on the surface of a planet:

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

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

The mass of a spherical planet can be expressed in terms of its volume and mean density ($\rho$) as:

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

Substituting the expression for mass ($M$) back into the formula for $g$:

$g=\frac{G}{R^2}\left(\frac{4}{3}\pi R^3\rho \right)=\frac{4}{3}\pi GR\rho$

Since $\frac{4}{3}\pi G$ is a constant, we can see that the acceleration due to gravity is directly proportional to both the radius ($R$) and the density ($\rho$) of the planet:

$g\propto R\rho$

Since the radius is half of the diameter ($D$), the ratio of the radii is equal to the ratio of the diameters:

$\frac{R_1}{R_2}$

We are given:
Ratio of diameters, $\frac{D_1}{D_2}$
Ratio of mean densities, $\frac{{\rho }_1}{{\rho }_2}$

Now, we can find the ratio of their accelerations due to gravity ($\frac{g_1}{g_2}$):

$\frac{g_1}{g_2}$

Substitute the given values into the equation:

$\frac{g_1}{g_2}$

Therefore, the acceleration due to gravity on the planets will be in the ratio of 2:1.