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_(2) gt W_(1) gt 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 weights of an object at different locations (coal mine, sea level, and top of a mountain), we will analyze how the gravitational force (weight) changes based on the distance from the center of the Earth.

### Step-by-Step Solution:

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

2. **Weight at Sea Level (W2)**:
At sea level, the weight is simply:
\[ W_2 = m \cdot g \]
Here, \( g \) is the standard acceleration due to gravity at sea level.

3. **Weight in the Coal Mine (W1)**:
When we go down into a coal mine, we are moving below the surface of the Earth. The variation of gravitational acceleration at a depth \( d \) is given by:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
where \( R \) is the radius of the Earth. Thus, the weight at this depth is:
\[ W_1 = m \cdot g' = m \cdot g \left(1 - \frac{d}{R}\right) \]

4. **Weight at the Top of the Mountain (W3)**:
When we go to a height \( h \) above sea level, the variation of gravitational acceleration is given by:
\[ g' = g \left(1 - \frac{2h}{R}\right) \]
Therefore, the weight at this height is:
\[ W_3 = m \cdot g' = m \cdot g \left(1 - \frac{2h}{R}\right) \]

5. **Comparing Weights**:
Now we need to compare \( W_1 \), \( W_2 \), and \( W_3 \):
- At sea level, \( W_2 = m \cdot g \).
- In the coal mine, \( W_1 = m \cdot g \left(1 - \frac{d}{R}\right) \) which is less than \( W_2 \) since \( d/R \) is positive.
- At the top of the mountain, \( W_3 = m \cdot g \left(1 - \frac{2h}{R}\right) \) which is also less than \( W_2 \) and generally less than \( W_1 \) if \( h \) is sufficiently large.

6. **Conclusion**:
From the analysis, we can conclude:
\[ W_2 > W_1 > W_3 \]
Therefore, the relationship between the weights is:
\[ W_2 > W_1 > W_3 \]

### Final Answer:
The correct relationship is \( W_2 > W_1 > W_3 \).

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