The radii of two planets are `R_1 and R_2` ans their densities are `rho_1 and rho_2` respectively. If `g_1 and g_2` represent surfaces , then `g_1/g_2` is

Options

A. `(rho_1R_2^2)/(rho_1R_1^2)`
B. `(rho_1R_2^1)/(rho_1R_2^2)`
C. `(rho_2R_1)/(rho_1R_2)`
D. `(rho_1R_1)/(rho_2R_2)`

Answer

To solve the problem, we need to find the ratio of the acceleration due to gravity on the surfaces of two planets, given their radii and densities.

### Step-by-Step Solution:

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

2. **Expressing Mass in Terms of Density**:
The mass \( M \) of a planet can be expressed in terms of its density \( \rho \) and volume \( V \):
\[ M = \rho \cdot V \]
The volume \( V \) of a sphere (which we assume the planets are) is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Thus, the mass can be rewritten as:
\[ M = \rho \cdot \frac{4}{3} \pi R^3 \]

3. **Substituting Mass in the Gravity Formula**:
For planet 1:
\[ g_1 = \frac{G \cdot M_1}{R_1^2} = \frac{G \cdot \left( \rho_1 \cdot \frac{4}{3} \pi R_1^3 \right)}{R_1^2} = \frac{G \cdot \rho_1 \cdot \frac{4}{3} \pi R_1^3}{R_1^2} = \frac{4}{3} \pi G \cdot \rho_1 \cdot R_1 \]

For planet 2:
\[ g_2 = \frac{G \cdot M_2}{R_2^2} = \frac{G \cdot \left( \rho_2 \cdot \frac{4}{3} \pi R_2^3 \right)}{R_2^2} = \frac{G \cdot \rho_2 \cdot \frac{4}{3} \pi R_2^3}{R_2^2} = \frac{4}{3} \pi G \cdot \rho_2 \cdot R_2 \]

4. **Finding the Ratio \( \frac{g_1}{g_2} \)**:
Now we can find the ratio of \( g_1 \) to \( g_2 \):
\[ \frac{g_1}{g_2} = \frac{\frac{4}{3} \pi G \cdot \rho_1 \cdot R_1}{\frac{4}{3} \pi G \cdot \rho_2 \cdot R_2} \]
The \( \frac{4}{3} \pi G \) terms cancel out:
\[ \frac{g_1}{g_2} = \frac{\rho_1 \cdot R_1}{\rho_2 \cdot R_2} \]

5. **Final Result**:
Thus, the ratio of the accelerations due to gravity on the surfaces of the two planets is:
\[ \frac{g_1}{g_2} = \frac{\rho_1 R_1}{\rho_2 R_2} \]