Two planets A and B have the same average density . Their radii `R_A and R_B` are such that `R_A: R_B = 3 : 1` . If `g_A and g_B` are the acceleration due to gravity at the surface of the planets , the `g_A : g_B` equals

Options

A. `3:1`
B. `1:3`
C. `1:9`
D. `sqrt3:1`

Answer

To solve the problem, we need to find the ratio of the acceleration due to gravity at the surfaces of two planets A and B, given that they have the same average density and their radii are in the ratio \( R_A : R_B = 3 : 1 \).

### Step-by-Step Solution:

1. **Understand the Formula for Acceleration due to Gravity**:
The acceleration due to gravity \( g \) at the surface of a planet is given by the formula:
\[ g = \frac{GM}{R^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

2. **Express Mass in Terms of Density**:
The mass \( M \) of a planet can be expressed in terms of its density \( \rho \) and volume \( V \):
\[ M = \rho V \]
For a spherical planet, the volume \( V \) is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Therefore, the mass can be rewritten as:
\[ M = \rho \left(\frac{4}{3} \pi R^3\right) \]

3. **Substitute Mass into the Gravity Formula**:
Substituting the expression for mass into the formula for \( g \):
\[ g = \frac{G \left(\rho \frac{4}{3} \pi R^3\right)}{R^2} \]
Simplifying this gives:
\[ g = \frac{4}{3} \pi G \rho R \]
This shows that \( g \) is directly proportional to the radius \( R \) when the density \( \rho \) is constant.

4. **Calculate the Ratio of Acceleration due to Gravity**:
Now, we can find the ratio of the accelerations due to gravity for planets A and B:
\[ \frac{g_A}{g_B} = \frac{\frac{4}{3} \pi G \rho R_A}{\frac{4}{3} \pi G \rho R_B} = \frac{R_A}{R_B} \]
Given that \( R_A : R_B = 3 : 1 \), we can write:
\[ \frac{g_A}{g_B} = \frac{3}{1} \]

5. **Final Result**:
Therefore, the ratio of the accelerations due to gravity at the surfaces of planets A and B is:
\[ g_A : g_B = 3 : 1 \]

### Conclusion:
The final answer is \( g_A : g_B = 3 : 1 \).