If the distance between earth and sun is increased by 2 % , then find percentage change in gravitational force acting between them .

Explanation

To solve the problem of finding the percentage change in gravitational force when the distance between the Earth and the Sun is increased by 2%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Gravitational Force Formula**:
The gravitational force \( F \) between two masses \( M_1 \) (mass of Earth) and \( M_2 \) (mass of Sun) separated by a distance \( R \) is given by:
\[ F = \frac{G M_1 M_2}{R^2} \]
where \( G \) is the universal gravitational constant.

2. **Identify the Change in Distance**:
The problem states that the distance \( R \) is increased by 2%. This means the new distance \( R' \) can be expressed as:
\[ R' = R + 0.02R = 1.02R \]

3. **Calculate the New Gravitational Force**:
The new gravitational force \( F' \) at the new distance \( R' \) is:
\[ F' = \frac{G M_1 M_2}{(R')^2} = \frac{G M_1 M_2}{(1.02R)^2} = \frac{G M_1 M_2}{1.0404 R^2} \]

4. **Relate the New Force to the Original Force**:
We can express the new force \( F' \) in terms of the original force \( F \):
\[ F' = \frac{F}{1.0404} \]

5. **Calculate the Change in Force**:
The change in force \( \Delta F \) can be calculated as:
\[ \Delta F = F' - F = \frac{F}{1.0404} - F = F \left(\frac{1}{1.0404} - 1\right) \]
Simplifying this gives:
\[ \Delta F = F \left(\frac{1 - 1.0404}{1.0404}\right) = F \left(-\frac{0.0404}{1.0404}\right) \]

6. **Calculate the Percentage Change in Force**:
The percentage change in force is given by:
\[ \text{Percentage Change} = \frac{\Delta F}{F} \times 100 = \left(-\frac{0.0404}{1.0404}\right) \times 100 \]
This simplifies to approximately:
\[ \text{Percentage Change} \approx -3.88\% \]
Rounding this, we can say the gravitational force decreases by approximately 4%.

### Final Answer:
The percentage change in gravitational force acting between the Earth and the Sun when the distance is increased by 2% is approximately **4% decrease**.