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 \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by Newton's law of gravitation:
\[ F = \frac{G m_1 m_2}{r^2} \]
where \( G \) is the universal gravitational constant.

2. **Define Initial Conditions**:
Let:
- \( m_E \) = mass of the Earth
- \( m_S \) = mass of the Sun
- \( r \) = initial distance between the Earth and the Sun
- The initial gravitational force \( F_1 \) is:
\[ F_1 = \frac{G m_E m_S}{r^2} \]

3. **Calculate the New Distance**:
If the distance is increased by 2%, the new distance \( r_2 \) can be calculated as:
\[ r_2 = r + 0.02r = 1.02r \]

4. **Calculate the New Gravitational Force**:
The new gravitational force \( F_2 \) at the new distance \( r_2 \) is:
\[ F_2 = \frac{G m_E m_S}{(1.02r)^2} \]
Simplifying this gives:
\[ F_2 = \frac{G m_E m_S}{1.0404 r^2} = \frac{F_1}{1.0404} \]

5. **Calculate the Change in Gravitational Force**:
The change in gravitational force \( \Delta F \) is:
\[ \Delta F = F_2 - F_1 = \frac{F_1}{1.0404} - F_1 \]
Factoring out \( F_1 \):
\[ \Delta F = F_1 \left( \frac{1}{1.0404} - 1 \right) \]
This simplifies to:
\[ \Delta F = F_1 \left( \frac{1 - 1.0404}{1.0404} \right) = F_1 \left( \frac{-0.0404}{1.0404} \right) \]

6. **Calculate the Percentage Change**:
The percentage change in gravitational force is given by:
\[ \text{Percentage Change} = \left( \frac{\Delta F}{F_1} \right) \times 100\% \]
Substituting for \( \Delta F \):
\[ \text{Percentage Change} = \left( \frac{-0.0404}{1.0404} \right) \times 100\% \]
Calculating this gives approximately:
\[ \text{Percentage Change} \approx -3.89\% \]

7. **Final Result**:
Thus, the percentage change in gravitational force is approximately:
\[ \text{Percentage Change} \approx -3.89\% \]
This indicates a decrease in gravitational force by about 3.89%, which can be rounded to approximately 4%.