The value of ‘ g ’ at a particular point is `9.8 m//s^(2)` . Suppose the earth suddenly shrinks uniformly to half its present size without losing any mass. The value of ‘ g ’ at the same point (assuming that the distance of the point from the centre of earth does not shrink) will now be

Options

A. `4.9 m//sec^(2)`
B. `3.1 m//sec^(2)`
C. `9.8 m//sec^(2)`
D. `19.6 m//sec^(2)`

Answer

To solve the problem, we need to understand how the gravitational acceleration \( g \) is affected by changes in the Earth's size while keeping its mass constant.

### Step-by-Step Solution:

1. **Understanding the formula for gravitational acceleration**:
The gravitational acceleration \( g \) at a distance \( r \) from the center of a mass \( M \) is given by the formula:
\[ g = \frac{GM}{r^2} \]
where \( G \) is the gravitational constant.

2. **Initial conditions**:
We are given that the value of \( g \) at a particular point is \( 9.8 \, \text{m/s}^2 \). This is calculated using the radius of the Earth \( R \) and the mass of the Earth \( M \).

3. **Change in Earth's size**:
The problem states that the Earth shrinks uniformly to half its present size. This means the new radius \( r' \) is:
\[ r' = \frac{R}{2} \]

4. **Mass remains constant**:
The mass of the Earth remains the same, so \( M \) does not change.

5. **Calculating new gravitational acceleration**:
We can now calculate the new gravitational acceleration \( g' \) at the same point (which is now at a distance \( R \) from the center of the Earth):
\[ g' = \frac{GM}{(R)^2} \]
Since the point is still at distance \( R \) from the center of the Earth, we can see that:
\[ g' = \frac{GM}{R^2} \]

6. **Comparison of \( g \) and \( g' \)**:
The original gravitational acceleration \( g \) was calculated at the radius \( R \):
\[ g = \frac{GM}{R^2} \]
Since the mass \( M \) and the distance \( R \) from the center of the Earth have not changed, we find that:
\[ g' = g = 9.8 \, \text{m/s}^2 \]

### Conclusion:
Thus, the value of \( g \) at the same point after the Earth shrinks to half its size (while keeping mass constant) remains the same:
\[ g' = 9.8 \, \text{m/s}^2 \]