Planetary Force Ratios Explained

This question asks us to determine the ratio of the forces that two planets exert on each other. We are given information about the ratios of their masses and their diameters.

Given Information for Planets

We are provided with the following ratios:

• The ratio of the masses of the two planets is \(m_1 : m_2 = 1:7\).
• The ratio of the diameters of the two planets is \(d_1 : d_2 = 2:1\).

Newton's Laws for Force Analysis

To understand the forces between celestial bodies like planets, we rely on fundamental laws of physics:

1. Newton's Law of Universal Gravitation: This law states that every particle attracts every other particle in the universe 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. The mathematical expression for this force (F) is:

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

   Here, \(m_1\) and \(m_2\) are the masses of the two objects, r is the distance between their centers, and G is the universal gravitational constant.

2. Newton's Third Law of Motion: This law is crucial for understanding interactions. It states that for every action, there is an equal and opposite reaction. In the context of forces between two objects, the force exerted by the first object on the second is equal in magnitude and opposite in direction to the force exerted by the second object on the first.

Analyzing Mutual Forces Between Planets

Let \(F_{12}\) be the force exerted by planet 1 on planet 2, and \(F_{21}\) be the force exerted by planet 2 on planet 1. According to Newton's Law of Universal Gravitation:

• Force exerted by planet 1 on planet 2:

  \(F_{12} = G \frac{m_1 m_2}{r^2}\)

• Force exerted by planet 2 on planet 1:

  \(F_{21} = G \frac{m_2 m_1}{r^2}\)

In both equations, \(m_1 m_2\) represents the product of the masses, and \(r^2\) represents the square of the distance between their centers. These quantities are the same regardless of which planet is considered the source and which is the target of the force.

Role of Newton's Third Law in Forces

Newton's Third Law directly applies here. It states that the action force (\(F_{12}\)) and the reaction force (\(F_{21}\)) are always equal in magnitude and opposite in direction. Therefore:

\(|F_{12}| = |F_{21}|\)

The question asks for the ratio of the forces they exert on each other, which refers to the ratio of the magnitudes of these forces.

Ratio = \(\frac{|F_{12}|}{|F_{21}|}\)

Substituting the expressions for the forces:

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

Simplifying this expression, we find:

Ratio = 1

Diameter Ratio's Relevance to Force

The information about the ratio of the diameters (\(d_1 : d_2 = 2:1\)) is provided. However, this information is not needed to calculate the ratio of the forces the planets exert on each other. Newton's Third Law ensures that these mutual forces are always equal in magnitude. The diameter ratio would be relevant for other calculations, such as the gravitational acceleration at the surface of each planet, but not for the interaction force between them.

Calculation of Force Ratio

Based on Newton's Third Law, the magnitudes of the forces exerted by the two planets on each other are equal.

Therefore, the ratio of the forces is:

Force Ratio = 1:1

Conclusion on Planetary Forces

The ratio between the forces which the two planets exert on each other is 1:1. This outcome is a direct consequence of Newton's Third Law of Motion, which governs all interactions, including gravitational ones.