Understanding the Gravitational Force Between Earth and Moon

This question asks about the gravitational force acting between the Earth and the Moon. To understand this, we need to recall Newton's Law of Universal Gravitation and Newton's Third Law of Motion.

Newton's Law of Universal Gravitation

Sir Isaac Newton's Law of Universal Gravitation states that every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Mathematically, the magnitude of this gravitational force ($\(F\)$) between two objects with masses \(m_1\) and \(m_2\) separated by a distance \(r\) is given by:

\(F = G \frac{m_1 m_2}{r^2}\)

Here, \(G\) is the universal gravitational constant.

In the case of the Earth and the Moon, let \(m_E\) be the mass of the Earth and \(m_M\) be the mass of the Moon. Let \(r\) be the distance between the center of the Earth and the center of the Moon. The magnitude of the gravitational force exerted by the Earth on the Moon (\(F_{EM}\)) and the magnitude of the gravitational force exerted by the Moon on the Earth (\(F_{ME}\)) are given by:

• \(F_{EM} = G \frac{m_E m_M}{r^2}\)
• \(F_{ME} = G \frac{m_M m_E}{r^2}\)

Since multiplication is commutative (\(m_E m_M = m_M m_E\)), we can see that the magnitude of the force exerted by the Earth on the Moon is equal to the magnitude of the force exerted by the Moon on the Earth:

\(F_{EM} = F_{ME}\)

This confirms that the gravitational forces between the Earth and the Moon have equal magnitudes.

Applying Newton's Third Law to Earth-Moon Gravity

Newton's Third Law of Motion states that for every action, there is an equal and opposite reaction. When one object exerts a force on a second object, the second object simultaneously exerts a force of equal magnitude and opposite direction on the first object.

The gravitational force is an action-reaction pair. If the Earth exerts a gravitational force on the Moon, the Moon simultaneously exerts a gravitational force on the Earth. According to Newton's Third Law:

• The force of the Earth on the Moon and the force of the Moon on the Earth are an action-reaction pair.
• These two forces must be equal in magnitude.
• These two forces must be opposite in direction.

The force exerted by the Earth on the Moon pulls the Moon towards the Earth. The force exerted by the Moon on the Earth pulls the Earth towards the Moon. These forces act along the line joining the centers of the two objects but point in opposite directions.

Analyzing the Options

Let's examine each option based on our understanding of gravitational force and Newton's Third Law:

• Option 1: "The force of gravity of the Earth on the Moon is greater than the force of gravity of the Moon on the Earth." This statement is false. Newton's Law of Gravitation and Newton's Third Law confirm that the magnitudes of the forces are equal.
• Option 2: "The force of gravity of the Moon on the Earth is greater than the force of gravity of the Earth on the Moon." This statement is also false, for the same reasons as option 1. The forces have equal magnitudes.
• Option 3: "The force of gravity of the Earth on the Moon and of the Moon on the Earth are equal in magnitude and are in the same direction." This statement is partially true (equal magnitude) but false regarding the direction. The forces are always in opposite directions as they are attractive forces pulling the objects towards each other along the connecting line.
• Option 4: "The force of gravity of the Earth on the Moon and of the Moon on the Earth are equal in magnitude but are in opposite directions." This statement perfectly aligns with both Newton's Law of Universal Gravitation regarding magnitude and Newton's Third Law regarding magnitude and direction.

Therefore, the only true statement is that the forces are equal in magnitude and opposite in direction.

Revision Table: Key Concepts of Gravitational Force

Additional Information: Effects of Earth-Moon Gravitational Force

While the forces are equal in magnitude, their effects on the Earth and the Moon are different due to the difference in their masses. According to Newton's Second Law (\(F = ma\) or \(a = F/m\)), the acceleration produced by a force is inversely proportional to the mass of the object.

• The Moon experiences a gravitational force from Earth. Because the Moon's mass is smaller than Earth's mass, this force causes the Moon to accelerate significantly, keeping it in orbit around the Earth.
• The Earth experiences an equal magnitude gravitational force from the Moon. However, because the Earth's mass is much larger than the Moon's mass, the acceleration of the Earth due to the Moon's gravity is much smaller. This force is still significant enough to cause tides in Earth's oceans and solid body.

So, while the forces are equal and opposite, the resulting accelerations (and thus motions) are different because the masses are different.