The percentage change in the acceleration of the earth towards the Sun from a total eclipse of the Sun to the point where the Moon is on a side of earth directly opposite to the Sun is

Options

A. (a)`(M_(s))/(M_(m)) (r_(2))/(r_(1))xx100`
B. (b)`(M_(s))/(M_(m))((r_(2))/(r_(1)))xx100`
C. (c)`2((r_(1))/(r_(2)))^(2) (M_(m))/(M_(s))xx100`
D. (d)`((r_(1))/(r_(2)))^(2)(M_(m))/(M_(s))xx100`

Explanation

To solve the problem of finding the percentage change in the acceleration of the Earth towards the Sun during a total eclipse compared to when the Moon is on the opposite side of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Forces Involved**:
- During a total eclipse, the Moon is between the Earth and the Sun. The gravitational forces acting on the Earth are due to both the Sun and the Moon.
- When the Moon is on the opposite side of the Earth, the gravitational force from the Moon acts in the opposite direction to that of the Sun.

2. **Define the Forces**:
- Let \( F_1 \) be the total gravitational force on the Earth during the total eclipse:
\[ F_1 = \frac{G M_s M_e}{r_1^2} + \frac{G M_m M_e}{r_2^2} \]
- Let \( F_2 \) be the total gravitational force on the Earth when the Moon is on the opposite side:
\[ F_2 = \frac{G M_s M_e}{r_1^2} - \frac{G M_m M_e}{r_2^2} \]

3. **Calculate the Change in Force**:
- The change in force \( \Delta F \) is given by:
\[ \Delta F = F_1 - F_2 \]
- Substituting the expressions for \( F_1 \) and \( F_2 \):
\[ \Delta F = \left( \frac{G M_s M_e}{r_1^2} + \frac{G M_m M_e}{r_2^2} \right) - \left( \frac{G M_s M_e}{r_1^2} - \frac{G M_m M_e}{r_2^2} \right) \]
- Simplifying this gives:
\[ \Delta F = 2 \frac{G M_m M_e}{r_2^2} \]

4. **Calculate the Change in Acceleration**:
- The change in acceleration \( \Delta A \) can be calculated using Newton's second law \( F = m \cdot a \):
\[ \Delta A = \frac{\Delta F}{M_e} \]
- Substituting for \( \Delta F \):
\[ \Delta A = \frac{2 G M_m M_e}{r_2^2 M_e} = \frac{2 G M_m}{r_2^2} \]

5. **Calculate Average Acceleration**:
- The average acceleration \( A_{avg} \) towards the Sun when the Moon is not affecting the Earth is:
\[ A_{avg} = \frac{G M_s}{r_1^2} \]

6. **Calculate the Percentage Change in Acceleration**:
- The percentage change in acceleration is given by:
\[ \text{Percentage Change} = \frac{\Delta A}{A_{avg}} \times 100 \]
- Substituting the values:
\[ \text{Percentage Change} = \frac{\frac{2 G M_m}{r_2^2}}{\frac{G M_s}{r_1^2}} \times 100 \]
- Simplifying this gives:
\[ \text{Percentage Change} = \frac{2 r_1^2 M_m}{r_2^2 M_s} \times 100 \]

7. **Final Result**:
- This expression gives the percentage change in the acceleration of the Earth towards the Sun from a total eclipse to the point where the Moon is on the opposite side.