The weight of a body on the surface of earth is W When the density of the body is doubled by keeping its radius constant, its weight on the surface of the moon is found to be` W_m` then ` (W_m)/(W_e) = __________(g_e =6g_m)`

Options

A. ` 1 :1`
B. ` 1 : 3`
C. ` 3 : 1`
D. ` 6 : 1`

Explanation

To solve the problem, we need to understand the relationship between weight, mass, and gravitational acceleration. Let's break it down step by step.

### Step-by-Step Solution

1. **Understanding Weight on Earth**:
The weight of a body on the surface of the Earth (denoted as \( W_e \)) is given by the formula:
\[ W_e = M \cdot g_e \]
where \( M \) is the mass of the body and \( g_e \) is the acceleration due to gravity on Earth.

2. **Effect of Doubling Density**:
When the density of the body is doubled while keeping its radius constant, the volume of the body does not change. The mass of the body can be expressed as:
\[ M = \text{Density} \times \text{Volume} \]
If the original density is \( \rho \), then the new density is \( 2\rho \). Therefore, the new mass \( M' \) becomes:
\[ M' = 2\rho \cdot V \]
Since the volume \( V \) remains constant, we can express the new mass as:
\[ M' = 2M \]

3. **Weight on the Moon**:
The weight of the body on the surface of the Moon (denoted as \( W_m \)) can be calculated using the new mass:
\[ W_m = M' \cdot g_m \]
Substituting \( M' = 2M \):
\[ W_m = 2M \cdot g_m \]

4. **Relating Gravitational Accelerations**:
We know that the acceleration due to gravity on the Moon \( g_m \) is related to that on Earth \( g_e \) by the equation:
\[ g_e = 6g_m \quad \text{(given)} \]
Therefore, we can express \( g_m \) in terms of \( g_e \):
\[ g_m = \frac{g_e}{6} \]

5. **Substituting for \( g_m \)**:
Now, we substitute \( g_m \) into the equation for \( W_m \):
\[ W_m = 2M \cdot \left(\frac{g_e}{6}\right) \]
Simplifying this gives:
\[ W_m = \frac{2M \cdot g_e}{6} = \frac{M \cdot g_e}{3} \]

6. **Finding the Ratio \( \frac{W_m}{W_e} \)**:
Now, we can find the ratio of the weights:
\[ \frac{W_m}{W_e} = \frac{\frac{M \cdot g_e}{3}}{M \cdot g_e} = \frac{1}{3} \]

### Final Answer
Thus, the ratio \( \frac{W_m}{W_e} \) is:
\[ \frac{W_m}{W_e} = \frac{1}{3} \]