Problem 39 - Astronomy: At Play in the Cosmos

Question

By what factor would the gravitational force between Earth and the Moon be greater if the mass of each body were twice as great and the distance were half as great as they are today?

Step-by-Step Solution

Answer

The gravitational force would be 16 times greater.

Step 1: Understand the Gravitational Force Formula

The gravitational force between two masses is given by the equation: \( F = \frac{G \, m_1 \, m_2}{r^2} \) where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the two bodies, and \( r \) is the distance between their centers.

Step 2: Determine the Original Gravitational Force

Let the original masses of Earth and the Moon be \( m_E \) and \( m_M \), and the original distance between them be \( r \). The original gravitational force is \( F_0 = \frac{G \, m_E \, m_M}{r^2} \).

Step 3: Define the New Conditions

Under the new conditions, both the Earth's and the Moon's masses are doubled, so the new masses are \( 2m_E \) and \( 2m_M \). The new distance is half, so it becomes \( \frac{r}{2} \).

Step 4: Calculate the New Gravitational Force

Substitute the new values into the gravitational force formula: \[ F_{new} = \frac{G \, (2m_E) \, (2m_M)}{(\frac{r}{2})^2} = \frac{4G \, m_E \, m_M}{\frac{r^2}{4}} = \frac{16G \, m_E \, m_M}{r^2} \].

Step 5: Calculate the Factor Increase

The factor by which the gravitational force increases is the ratio of the new force to the original force: \[ \text{Factor} = \frac{F_{new}}{F_0} = \frac{16G \, m_E \, m_M / r^2}{G \, m_E \, m_M / r^2} = 16 \].

Key Concepts

Gravitational Constant

The gravitational constant, denoted by the letter \( G \), is a fundamental constant of nature that appears in the universal law of gravitation. It represents the strength of gravitational force exerted between two bodies. Mathematically, it is defined in the equation for gravitational force:

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

where \( F \) is the gravitational force, \( m_1 \) and \( m_2 \) are the masses of the two objects, and \( r \) is the distance between their centers. The value of \( G \) is approximately \( 6.674 \, \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \).

This constant helps us quantify the attractive force between any two bodies in the universe and shows us that gravity is relatively weak compared to other fundamental forces. Despite its weakness, given the massive scale of celestial bodies, gravitational force becomes the dominant force in shaping the motion of planets, stars, and galaxies.

Mass and Distance

The gravitational force between two objects is, in part, determined by their masses. Greater mass means a stronger gravitational pull. Let's consider the gravitational force formula:

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

As you can see, the force is directly proportional to the product of the two masses \( m_1 \) and \( m_2 \). Doubling either mass would double the gravitational force between them. Thus, if both masses are doubled, the force increases by a factor of four, as we multiply two by two.

Distance plays a critical role in gravitational interactions as well. The gravitational force is inversely proportional to the square of the distance between the centers of the two masses. So if the distance \( r \) is halved, the force increases. Specifically, halving the distance decreases the denominator by a factor of four \( (\frac{1}{r^2/4} = \frac{4}{r^2}) \), causing the gravity effect to intensify quadratically.

Inverse-Square Law

The inverse-square law is an essential principle in physics, particularly in the context of gravitational force. It helps us understand how the force between two objects changes as the distance between them changes.

• The law states that the force is inversely proportional to the square of the distance separating the objects.
• Expressed mathematically in the gravitational force equation: \( F \propto \frac{1}{r^2} \).

This principle implies that even a small change in distance can lead to a significant change in the gravitational force. For example, if you reduce the distance by half, the force does not merely double but increases by four times. This is because you square the factor by which the distance changed (i.e., \( 2^2 \)).

This law is crucial when examining celestial bodies, such as planets and moons, since the distances involved are vast. It ensures we account for the geometric nature of space and the spread of force fields, illustrating why even far-away objects can exert noticeable gravitational pulls.