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Chapter 5: Problem 34

Choose the best answer to each of the following. Explain your reasoning with one or more complete sentences. If Earth were twice as far from the Sun, the force of gravity attracting Earth to the Sun would be (a) twice as strong. (b) half as strong. (c) one-quarter as strong.

Short Answer

Expert verified
The force would be one-quarter as strong if the Earth were twice as far from the Sun, so the answer is (c).

Step by step solution

01

Identify the Relationship

The force of gravity between two masses is determined by Newton's law of universal gravitation. It states that the gravitational force \( F \) is proportional to the product of the two masses and inversely proportional to the square of the distance between their centers. The formula is \( F = \frac{G \, m_1 \, m_2}{r^2} \), where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses, and \( r \) is the distance between the masses.
02

Apply the Formula

In this scenario, we need to determine how the gravitational force changes if the distance between Earth and the Sun is doubled. By applying the formula, if the distance \( r \) is doubled, then the gravitational force \( F \) will be \( F = \frac{G \, m_1 \, m_2}{(2r)^2} \).
03

Simplify the Expression

Simplify the expression to understand the change in force: \[ F = \frac{G \, m_1 \, m_2}{4r^2} \]. This shows that the gravitational force would be one-quarter of its original value when the distance is doubled. The relationship between the distance and gravity is that doubling the distance will decrease the force by a factor of four.

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Key Concepts

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

Gravitational Force
Gravitational force is a fundamental force of nature that acts between any two masses. It is what keeps you on the ground and the planets in orbit around the Sun. This force is defined by Newton's law of universal gravitation. The formula given by Isaac Newton to calculate the gravitational force is:
\[ F = \frac{G \, m_1 \, m_2}{r^2} \]
Here:
  • \( F \) is the gravitational force between two objects.
  • \( G \) is the gravitational constant, a value that helps scale the size of the force appropriately.
  • \( m_1 \) and \( m_2 \) are the masses of the two objects.
  • \( r \) is the distance between the centers of the two masses.
The larger the mass of either object, the stronger the gravitational force. This explains why massive objects, like stars and planets, have significant gravitational pulls.
Inverse Square Law
The inverse square law is a key principle related to various forces and phenomena in physics, including gravity. According to this law, the force between two objects is inversely proportional to the square of the distance between them. This means:
If you double the distance between the two masses:
  • The gravitational force becomes \( \frac{1}{4} \) of its original value.
Mathematically, we see this illustrated in the formula for gravitational force: \( F = \frac{G \, m_1 \, m_2}{r^2} \).
The relationship can sound complex, but think of it this way:
  • As the distance increases, the force decreases rapidly - not linearly, but according to the square of the distance.
  • This rapid decrease highlights why the inverse square law is crucial for understanding gravity's effects over large distances, such as those between Earth and the Sun.
Distance and Gravity Relationship
The relationship between distance and gravity is central to many phenomena we observe in our universe. As you鈥檝e learned through the inverse square law, increasing the distance between two objects drastically reduces the gravitational force.
In practical terms:
  • If Earth were twice as far from the Sun, the gravitational pull it feels from the Sun would be only a quarter as strong as it is now.
This principle helps us understand planetary orbits and why planets closer to the Sun, like Mercury, experience much stronger gravitational attraction compared to those further away, such as Neptune.
Understanding this relationship is crucial in fields ranging from astronomy to physics, as it explains how celestial bodies interact over vast distances and the stability of their orbits.

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Most popular questions from this chapter

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