Two planets A and B have the same averge 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 surfaces of the planets, the `g_A : g_B` equals

Options

A. `3 : 1`
B. `1 : 3`
C. `1 : 9`
D. `sqrt(3) : 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. **Understanding the Formula for Acceleration due to Gravity**:
The acceleration due to gravity \( g \) at the surface of a planet can be expressed as:
\[ g = \frac{G \cdot M}{R^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

2. **Relating Mass to Density**:
The mass \( M \) of a planet can be expressed in terms of its volume and density:
\[ M = \rho \cdot V \]
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Therefore, we can rewrite the mass as:
\[ M = \rho \cdot \frac{4}{3} \pi R^3 \]

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

4. **Finding the Ratio of Accelerations**:
For planets A and B, we can write:
\[ g_A = \frac{4}{3} \pi G \rho R_A \]
\[ g_B = \frac{4}{3} \pi G \rho R_B \]
Taking the ratio \( \frac{g_A}{g_B} \):
\[ \frac{g_A}{g_B} = \frac{R_A}{R_B} \]

5. **Substituting the Given Ratio of Radii**:
We know from the problem statement that:
\[ \frac{R_A}{R_B} = \frac{3}{1} \]
Therefore:
\[ \frac{g_A}{g_B} = \frac{3}{1} \]

6. **Conclusion**:
The ratio of the acceleration due to gravity at the surfaces of planets A and B is:
\[ g_A : g_B = 3 : 1 \]

### Final Answer:
\[ g_A : g_B = 3 : 1 \]