The weight of a body on the surface of earth is `W_e.` 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 find the ratio \( \frac{W_m}{W_e} \) given the conditions about the body’s weight on Earth and the Moon.

### Step 1: Understand the relationship between weight, mass, and gravity
The weight of an object is given by the formula:
\[ W = m \cdot g \]
where \( W \) is the weight, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity.

### Step 2: Calculate the weight on Earth
Let the density of the body be \( \rho \) and the volume \( V \) of the body can be expressed in terms of its radius \( r \):
\[ V = \frac{4}{3} \pi r^3 \]
The mass \( m \) of the body can be expressed as:
\[ m = \rho \cdot V = \rho \cdot \frac{4}{3} \pi r^3 \]
Thus, the weight on Earth \( W_e \) is:
\[ W_e = m \cdot g_e = \left( \rho \cdot \frac{4}{3} \pi r^3 \right) \cdot g_e \]

### Step 3: Calculate the weight on the Moon after doubling the density
When the density is doubled, the new density becomes \( 2\rho \). The mass of the body now is:
\[ m' = 2\rho \cdot V = 2\rho \cdot \frac{4}{3} \pi r^3 \]
The weight on the Moon \( W_m \) is:
\[ W_m = m' \cdot g_m = \left( 2\rho \cdot \frac{4}{3} \pi r^3 \right) \cdot g_m \]

### Step 4: Substitute \( g_e \) in terms of \( g_m \)
We know from the problem statement that \( g_e = 6g_m \). Therefore, we can express \( W_e \) as:
\[ W_e = \left( \rho \cdot \frac{4}{3} \pi r^3 \right) \cdot 6g_m \]

### Step 5: Calculate the ratio \( \frac{W_m}{W_e} \)
Now we can substitute \( W_m \) and \( W_e \) into the ratio:
\[ \frac{W_m}{W_e} = \frac{2\rho \cdot \frac{4}{3} \pi r^3 \cdot g_m}{\rho \cdot \frac{4}{3} \pi r^3 \cdot 6g_m} \]
The \( \rho \), \( \frac{4}{3} \pi r^3 \), and \( g_m \) cancel out:
\[ \frac{W_m}{W_e} = \frac{2}{6} = \frac{1}{3} \]

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

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