If the mass of the earth remains constant but the diameter of the earth becomes two times its present value, then the weight of a person weighing 80 kg - wt, would be

Options

A. 20 kg - wt
B. 40 kg - wt
C. 60 kg - wt
D. 80 kg - wt

Explanation

To solve the problem step by step, we will analyze the effects of changing the diameter of the Earth on the weight of a person.

### Step-by-Step Solution:

1. **Understand the Concept of Weight**:
The weight \( W \) 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. **Initial Conditions**:
- The mass of the person is \( m = 80 \, \text{kg} \).
- The initial diameter of the Earth is \( D \), which means the radius \( r \) is \( \frac{D}{2} \).
- The acceleration due to gravity \( g \) at the surface of the Earth is given by:
\[ g = \frac{GM}{r^2} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

3. **New Conditions**:
- The diameter of the Earth becomes \( 2D \), which means the new radius \( r' \) is:
\[ r' = \frac{2D}{2} = D \]
- The mass of the Earth remains constant at \( M \).

4. **Calculate New Acceleration due to Gravity**:
The new acceleration due to gravity \( g' \) at the new radius \( r' \) is given by:
\[ g' = \frac{GM}{(r')^2} = \frac{GM}{D^2} \]

5. **Relate New Gravity to Old Gravity**:
Since the original radius was \( r = \frac{D}{2} \), we can express the original gravity \( g \) as:
\[ g = \frac{GM}{\left(\frac{D}{2}\right)^2} = \frac{GM}{\frac{D^2}{4}} = \frac{4GM}{D^2} \]
Thus, we can relate \( g' \) to \( g \):
\[ g' = \frac{GM}{D^2} = \frac{1}{4} \cdot g \]

6. **Calculate New Weight**:
The new weight \( W' \) of the person after the change in diameter is:
\[ W' = m \cdot g' = 80 \, \text{kg} \cdot \left(\frac{1}{4} g\right) = \frac{80 \, \text{kg} \cdot g}{4} = 20 \, \text{kg} \cdot g \]
Since weight is measured in Newtons, we can express the new weight as:
\[ W' = \frac{W}{4} \]
where \( W \) is the original weight.

7. **Conclusion**:
The new weight of the person after the diameter of the Earth is doubled is:
\[ W' = 20 \, \text{kg} \]

### Final Answer:
The weight of the person would be **20 kg**.

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