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The problem you're presenting involves calculating the gravitational force between the Earth and the Moon. The diagram provided illustrates the Earth and the Moon with their respective masses, and the distance between their centers of mass.

The given data is:

- Mass of the Earth, \( M_E = 5.97 \times 10^{24} \, \text{kg} \)
- Mass of the Moon, \( M_M = 7.35 \times 10^{22} \, \text{kg} \)
- Distance between Earth and Moon, \( r = 3.80 \times 10^8 \, \text{m} \)

The force of gravitational attraction between two objects is given by Newton's law of universal gravitation:

\[
F = \frac{G \cdot M_E \cdot M_M}{r^2}
\]

Where:
- \( G \) is the gravitational constant, \( G = 6.674 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \).

### Calculation:

1. Plug in the known values:
\[
F = \frac{(6.674 \times 10^{-11}) \cdot (5.97 \times 10^{24}) \cdot (7.35 \times 10^{22})}{(3.80 \times 10^8)^2}
\]

2. Simplifying:
\[
F = \frac{(6.674 \times 10^{-11}) \cdot (5.97 \times 10^{24}) \cdot (7.35 \times 10^{22})}{(1.444 \times 10^{17})}
\]

3. Finally:
\[
F \approx 1.98 \times 10^{20} \, \text{N}
\]

### Result:
The force exerted by Earth on the Moon and vice versa is approximately \( F \approx 1.98 \times 10^{20} \, \text{N} \).

Since force vectors are opposite but equal in magnitude (according to Newton’s Third Law), the force vectors can be expressed as:
- Force on Moon by Earth: \( \vec{F}_{ME} = (1.98 \times 10^{20} \, \text{N}) \hat{i} \)
- Force on Earth by Moon: \( \vec{F}_{EM} = (-1.98 \times 10^{20} \, \text{N}) \hat{i} \)

Would you like me to go into further details, or do you have any specific questions?

Here are 5 related questions to explore further:
1. What is Newton's Third Law of Motion, and how does it apply in this scenario?
2. How would the force change if the distance between Earth and Moon increased?
3. How does the gravitational force affect the orbital motion of the Moon around the Earth?
4. Can we calculate the acceleration experienced by the Earth and Moon due to this gravitational force?
5. How does the gravitational force between two objects change with mass?

**Tip:** Gravitational force decreases with the square of the distance between two objects. Hence, even small changes in distance can significantly affect the force.

Calculate the force exerted by Earth on the Moon and the force exerted by the Moon on Earth. The mass of Earth is 5.97 × 10^24 kg, the mass of the Moon is 7.35 × 10^22 kg, and the distance between their centers of mass is 3.80 × 10^8 m.

This problem demonstrates how to calculate the gravitational force between Earth and the Moon using Newton's law of universal gravitation. With masses of 5.97 × 10^24 kg and 7.35 × 10^22 kg, and a distance of 3.80 × 10^8 m between them, learn step-by-step how to solve for the gravitational force.

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