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Chapter 4: Problem 20

If the Moon were 2 times closer to Earth than it is now, the gravitational force between Earth and the Moon would be a. 2 times stronger. b. 4 times stronger. a. 2 times stronger. b. 4 times stronger.

Short Answer

Expert verified
b. 4 times stronger.

Step by step solution

01

Identify the formula

The gravitational force between two masses is given by Newton's Law of Gravitation:
02

Compare distances

When the distance between two objects is halved (2 times closer), we calculate the new force by determining the ratio of the squares of the distances.
03

Gravitational force relation

According to Newton's law, the gravitational force is inversely proportional to the square of the distance:
04

Substitute values

Now we substitute the new distance into the formula:
05

Calculate new force

The new force can be written as:

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gravitational Force
The gravitational force between two objects is a fundamental concept in physics. It comes from Newton’s Law of Gravitation and describes how two masses attract each other. This force is given by the equation:
\[ F = \frac{G \, m_1 \, m_2}{r^2} \]
Here:
  • \( F \) is the gravitational force.
  • \( G \) is the gravitational constant.
  • \( m_1 \) and \( m_2 \) are the masses of the two objects.
  • \( r \) is the distance between the centers of the two masses.
Gravitational force is always attractive and acts along the line joining the centers of the two objects.
Inverse Square Law
Newton's Law of Gravitation follows the inverse square law. This law states that the gravitational force between two objects is inversely proportional to the square of the distance between them.

Mathematically, it can be expressed as: \[ F \propto \frac{1}{r^2} \]
This means:
  • If the distance \( r \) between the objects is doubled, the gravitational force becomes one-fourth.
  • If the distance is halved, the force becomes four times stronger.
The inverse square law is crucial in understanding how distance affects the strength of forces like gravity.
Distance and Force Relationship
The relationship between distance and gravitational force is a key concept in this exercise.
  • If the Moon were twice as close to Earth as it is now, we need to apply the inverse square law to understand the change in gravitational force.
Since the distance \( r \) is halved (making it \( \frac{1}{2} \) of the original distance), we substitute \( r \rightarrow \frac{r}{2} \) into the inverse square formula:
\[ \frac{1}{\left(\frac{r}{2}\right)^2} = \frac{1}{\frac{r^2}{4}} = 4 \]
This calculation shows that the gravitational force would be \textbf{4 times stronger} if the Moon were twice as close to Earth. It highlights how the inverse square law can be practically applied to predict changes in gravitational force based on changes in distance.

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