If the earth shrinks to half of its radius and mass remains constant, then the weight of an object on earth will become

Options

A. doubled
B. halved
C. fourtimes
D. same

Answer

To solve the problem, we need to analyze how the weight of an object changes when the radius of the Earth is halved while its mass remains constant.

### Step-by-Step Solution:

1. **Understanding Weight**:
The weight \( W \) of an object on the surface of the Earth 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 at the surface of the Earth.

2. **Acceleration Due to Gravity**:
The acceleration due to gravity \( g \) at the surface of the Earth can be expressed using the formula:
\[ g = \frac{G \cdot M}{R^2} \]
where \( G \) is the universal gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

3. **New Radius**:
If the Earth shrinks to half of its radius, the new radius \( R' \) is:
\[ R' = \frac{R}{2} \]

4. **Calculating New Acceleration Due to Gravity**:
We can find the new acceleration due to gravity \( g' \) at the surface of the Earth after it has shrunk:
\[ g' = \frac{G \cdot M}{(R')^2} = \frac{G \cdot M}{\left(\frac{R}{2}\right)^2} = \frac{G \cdot M}{\frac{R^2}{4}} = 4 \cdot \frac{G \cdot M}{R^2} = 4g \]

5. **Calculating New Weight**:
The new weight \( W' \) of the object when the radius is halved is:
\[ W' = m \cdot g' = m \cdot (4g) = 4 \cdot (m \cdot g) = 4W \]
Thus, the weight of the object becomes four times its original weight.

6. **Conclusion**:
Therefore, if the Earth shrinks to half of its radius while its mass remains constant, the weight of an object on Earth will become four times its original weight.

### Final Answer:
The weight of an object on Earth will become \( 4 \) times its original weight.

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