The densities of two planets are in the ratio of 2 : 3 and their radii are in the ratio of 1 : 2. What is the ratio of acceleration due to gravity at their surfaces ?

Options

A. `1 : 3`
B. `3 : 1`
C. `1 : 9`
D. `9 : 4`

Explanation

To find the ratio of acceleration due to gravity at the surfaces of two planets, we can use the formula for gravitational acceleration:

\[ g = \frac{GM}{R^2} \]

where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet. The mass \( M \) can be expressed in terms of density \( \rho \) and volume \( V \):

\[ M = \rho V \]

The volume \( V \) of a sphere is given by:

\[ V = \frac{4}{3} \pi R^3 \]

Thus, the mass can be rewritten as:

\[ M = \rho \left(\frac{4}{3} \pi R^3\right) \]

Substituting this into the formula for \( g \):

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

Now, let's denote the two planets as Planet 1 and Planet 2. The densities and radii of the two planets are given as follows:

- Density ratio: \( \frac{d_1}{d_2} = \frac{2}{3} \)
- Radius ratio: \( \frac{r_1}{r_2} = \frac{1}{2} \)

We need to find the ratio of the acceleration due to gravity at their surfaces, \( \frac{g_1}{g_2} \):

\[ g_1 = \frac{4}{3} \pi G d_1 r_1 \]
\[ g_2 = \frac{4}{3} \pi G d_2 r_2 \]

Taking the ratio \( \frac{g_1}{g_2} \):

\[ \frac{g_1}{g_2} = \frac{d_1 r_1}{d_2 r_2} \]

Substituting the ratios of density and radius:

\[ \frac{g_1}{g_2} = \frac{d_1}{d_2} \cdot \frac{r_1}{r_2} = \frac{\frac{2}{3}}{\frac{3}{3}} \cdot \frac{\frac{1}{2}}{1} = \frac{2}{3} \cdot \frac{1}{2} = \frac{2}{6} = \frac{1}{3} \]

Thus, the ratio of the acceleration due to gravity at the surfaces of the two planets is:

\[ \frac{g_1}{g_2} = \frac{1}{3} \]