The weight of an object in the coal mine, sea level and at the top of the mountain are `W_(1),W_(2)` and `W_(3)` respectively, then

Options

A. `w_(1) lt w_(2) lt w_(3)`
B. `w_(1) = w_(2) = w_(3)`
C. `w_(1) lt w_(2) lt w_(3)`
D. `w_(1) gt w_(2) gt w_(3)`

Explanation

To solve the problem regarding the weight of an object in a coal mine, at sea level, and at the top of a mountain, we need to understand how gravitational force (weight) varies with altitude and depth.

### Step-by-Step Solution:

1. **Understanding Weight**:
The weight of an object is given by the formula:
\[ W = mg \]
where \( W \) is the weight, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity.

2. **Weight at Sea Level (W2)**:
At sea level, the value of \( g \) is approximately \( 9.81 \, \text{m/s}^2 \). Therefore, the weight of the object at sea level is:
\[ W_2 = mg \]

3. **Weight in a Coal Mine (W1)**:
When we go deep into a coal mine, we are moving below the Earth's surface. The value of \( g \) decreases slightly as we go deeper due to the reduction in the effective mass of the Earth above us. Thus, the weight in the coal mine is:
\[ W_1 = m g_1 \quad \text{(where \( g_1 < g \))} \]
This means \( W_1 < W_2 \).

4. **Weight at the Top of a Mountain (W3)**:
At the top of a mountain, we are at a higher altitude, which also causes the value of \( g \) to decrease. Therefore, the weight at the top of the mountain is:
\[ W_3 = m g_3 \quad \text{(where \( g_3 < g \))} \]
This means \( W_3 < W_2 \).

5. **Comparing the Weights**:
From the above analysis, we can conclude:
- \( W_1 < W_2 \) (Weight in coal mine is less than at sea level)
- \( W_3 < W_2 \) (Weight at the top of the mountain is less than at sea level)

6. **Final Conclusion**:
Since both \( W_1 \) and \( W_3 \) are less than \( W_2 \), we can summarize the relationship as:
\[ W_1 < W_2 > W_3 \]