Question

The ratio of the mass of the Earth to the mass of the Moon is \(81 .\) The magnitude of the gravitational force that the Earth exerts on the Moon is a) the same as the magnitude of the gravitational force that the Moon exerts on the Earth. b) 81 times the gravitational force exerted by the Moon on the Earth. c) \(1 / 81\) of the gravitational force exerted by the Moon on the Earth. d) 9 times the gravitational force exerted by the Moon on the Earth. e) \(1 / 9\) of the gravitational force exerted by the Moon on the Earth.

Short answer

Answer: The magnitude of the gravitational force that the Earth exerts on the Moon is the same as the magnitude of the gravitational force that the Moon exerts on the Earth.

Step-by-step solution

Write the formula for the gravitational force

The formula for the gravitational force between two objects is given by: $$F = G\frac{m_1m_2}{r^2}$$ where \(F\) is the gravitational force, \(G\) is the gravitational constant, \(m_1\) and \(m_2\) are the masses of the two objects, and \(r\) is the distance between them.

Apply Newton's Third Law

Newton's Third Law states that for every action, there is an equal and opposite reaction. In this case, the gravitational force exerted by the Earth on the Moon is the action, and the gravitational force exerted by the Moon on the Earth is the reaction. So we can say that the magnitudes of these forces are equal, represented as: $$F_{Earth} = F_{Moon}$$

Use the ratio of the masses

We're given the ratio of the mass of the Earth to the mass of the Moon as 81. Let \(m_{Earth} = 81m_{Moon}\).

Substitute the equation for the gravitational force

Applying the gravitational force equation to the Earth and the Moon, we have: $$G\frac{m_{Earth}m_{Moon}}{r^2} = G\frac{m_{Moon}m_{Earth}}{r^2}$$

Substitute the mass ratio

Now, substitute the mass ratio \(m_{Earth} = 81m_{Moon}\): $$G\frac{(81m_{Moon})(m_{Moon})}{r^2} = G\frac{(m_{Moon})(m_{Earth})}{r^2}$$

Simplify the equation

Cancel out the gravitational constant \(G\) on both sides and divide both sides by \((m_{Moon})(m_{Earth})\): $$\frac{81m_{Moon}^2}{m_{Earth}m_{Moon}} = \frac{m_{Earth}m_{Moon}}{m_{Earth}m_{Moon}}$$ Simplify further to get: $$81 = 1$$

Analyze the results

Since the equation \(81 = 1\) is not true, this implies that the magnitudes of the gravitational forces exerted by the Earth and the Moon on each other must be equal. This is consistent with Newton's Third Law, where the action and reaction forces have equal magnitudes.

Choose the correct option

Based on the analysis, the correct answer is: a) the same as the magnitude of the gravitational force that the Moon exerts on the Earth.

Key concepts

Newton's Third Law

Understanding Newton's Third Law is crucial when discussing gravitational forces. This fundamental principle of physics states that for every action, there is an equal and opposite reaction. But what does this mean in the context of celestial bodies like the Earth and the Moon?

Simply put, if the Earth exerts a gravitational pull on the Moon, the Moon simultaneously exerts an equal force in the opposite direction on the Earth. This is why astronauts in space experience weightlessness; it's not that there's no gravity, but rather that the Earth's pull is met with an equal force from their spacecraft. These action-reaction force pairs are present in all interactions and not limited by the objects' masses. So, regardless of their size difference, the Earth and the Moon exert equal gravitational forces on each other.

Gravitational Constant

and it describes the strength of gravity in the context of the force between two masses. Think of it as the universe’s dial for gravity: if you could turn it up, objects would attract each other more strongly.

To calculate the gravitational force two objects exert on one another, we multiply the gravitational constant by the masses of the objects and divide by the square of the distance between their centers. The constancy of G means that the gravitational attraction between any two objects is predictable no matter where or when in the universe you are.

Mass Ratio

The mass ratio of two objects, like Earth and the Moon, is a comparison of their masses. It's an expression of how many times heavier or lighter one object is compared to another. When we say the Earth's mass is 81 times that of the Moon, we're providing a ratio that helps us understand the relative sizes of these celestial bodies.

However, it's important to note that while mass ratio can give us an idea of the relative size, it doesn’t directly tell us how strong the gravitational force will be. The gravitational force depends on both mass and distance as described by the universal law of gravitation. So, one should not make the mistake of assuming that a larger mass ratio means a stronger gravitational pull without considering the distances involved.

Action-Reaction Forces

Diving deeper into action-reaction forces can offer more insights beyond the basic statement of Newton's Third Law. These forces are part of a myriad of interactions, and in our case, they play a starring role in the dance between the Earth and the Moon. When analyzing gravitational pull, it's apparent that these forces are a perfect example of the action-reaction principle.

While the Earth may be significantly more massive, this doesn't change the fact that the force it experiences from the Moon's gravity is just as powerful as the force it exerts on the Moon. Remember, forces come in pairs; this equality is true whether you're talking about planets or playing tug-of-war. Acknowledging the equality of these forces helps to shatter common misconceptions that the larger body 'dominates' the gravitational interaction.