Suppose there are two planets, 1 and 2, having the same density but their radii are `R_(1) and R_(2)` respectively, where `R_(1) gt R_(2).` The accelerations due to gravity on the surface of these planets are related as

Options

A. `g_(1)gt g_(2)`
B. `g_(1)lt g_(2)`
C. `g_(1)=g_(2)`
D. Can't say anything

Explanation

To solve the problem of comparing the accelerations due to gravity on the surfaces of two planets with the same density but different radii, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Formula for Acceleration Due to Gravity**:
The acceleration due to gravity \( g \) on the surface of a planet is given by the formula:
\[ g = \frac{G M}{R^2} \]
where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) 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 \( V \):
\[ M = \rho V \]
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. **Substituting 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:
\[ g = \frac{4}{3} \pi G \rho R \]

4. **Comparing the Accelerations on Both Planets**:
Let’s denote the accelerations due to gravity on planets 1 and 2 as \( g_1 \) and \( g_2 \) respectively. Then we have:
\[ g_1 = \frac{4}{3} \pi G \rho R_1 \]
\[ g_2 = \frac{4}{3} \pi G \rho R_2 \]

5. **Finding the Relationship Between \( g_1 \) and \( g_2 \)**:
Since both planets have the same density \( \rho \) and the same constant factors \( \frac{4}{3} \pi G \), we can compare \( g_1 \) and \( g_2 \):
\[ \frac{g_1}{g_2} = \frac{R_1}{R_2} \]
Given that \( R_1 > R_2 \), it follows that:
\[ g_1 > g_2 \]

### Conclusion:
Thus, the relationship between the accelerations due to gravity on the surfaces of the two planets is:
\[ g_1 > g_2 \]