Question

Newton's Universal Law of Gravitation a. How does quadrupling the distance between two objects affect the gravitational force between them?b. Suppose the Sun were somehow replaced by a star with twice as much mass. What would happen to the gravitational force between Earth and the Sun? c. Suppose Earth were moved to one-third of its current distance from the Sun. What would happen to the gravitational force between Earth and the Sun?

Short answer

a. The force becomes 1/16th. b. The force doubles. c. The force becomes nine times greater.

Step-by-step solution

Understand the Universal Law of Gravitation

The Universal Law of Gravitation is given by the formula: \( F = \frac{G \cdot m_1 \cdot m_2}{r^2} \), 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 the centers of the two masses.

Effect of Quadrupling the Distance on Gravitational Force

According to the formula, if the distance \( r \) is quadrupled (i.e., \( 4r \)), then the new gravitational force \( F' = \frac{G \cdot m_1 \cdot m_2}{(4r)^2} = \frac{G \cdot m_1 \cdot m_2}{16r^2} \). Thus, the gravitational force becomes \( \frac{1}{16} \) of its original value.

Effect of Doubling the Mass of the Sun on Gravitational Force

If the Sun's mass \( m_2 \) is doubled, the new gravitational force \( F'' = \frac{G \cdot m_1 \cdot 2m_2}{r^2} = 2 \cdot \frac{G \cdot m_1 \cdot m_2}{r^2} \). Thus, the gravitational force between the Earth and the Sun would double.

Effect of Moving Earth to One-Third its Current Distance from the Sun

Moving Earth to one-third of its current distance \( \frac{r}{3} \) means the new gravitational force \( F''' = \frac{G \cdot m_1 \cdot m_2}{(\frac{r}{3})^2} = \frac{G \cdot m_1 \cdot m_2}{\frac{r^2}{9}} = 9 \cdot \frac{G \cdot m_1 \cdot m_2}{r^2} \). Thus, the gravitational force would be nine times greater.

Key concepts

Newton's Laws

Newton's Laws, particularly the Universal Law of Gravitation, lay the foundation for understanding gravitational interactions. Sir Isaac Newton formulated these laws to describe the motion of objects and how they interact with one another. The Universal Law of Gravitation states that every point mass in the universe attracts every other point mass 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. This understanding allows us to predict how objects will behave under the influence of gravity, whether they are on Earth or in outer space. By grasping Newton's Laws, students can solve complex problems related to gravitational force with clarity and precision.
The formula: \[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \] helps quantify this interaction. Here, \(F\) represents the gravitational force, \(G\) is the gravitational constant, \(m_1\) and \(m_2\) are the masses of the objects involved, and \(r\) is the distance between them.

gravitational force

Gravitational force is a fundamental concept that explains the attraction between two masses. It is a mutual force that exists between any two objects, regardless of size or distance. The gravitational force acts along the line joining the centers of two masses. It is always attractive, pulling the masses towards each other.
Key aspects of gravitational force include:

• It increases with larger mass: The larger the masses of the objects, the stronger the gravitational force between them.
• It decreases with greater distance: As the distance between two objects increases, the gravitational force diminishes since it is inversely proportional to the square of the distance \((r)\).

The gravitational force is never zero where masses exist but becomes negligible over very large distances.

distance and gravity

The relationship between distance and gravity is crucial in astronomical calculations. According to Newton's Universal Law of Gravitation, gravitational force reduces significantly as the distance between two objects increases. This is because the force is inversely proportional to the square of the distance.
For instance:

• If the distance between two masses is doubled, the gravitational force becomes one-fourth (\(\frac{1}{4}\)) of the original.
• If the distance is tripled, the force becomes one-ninth (\(\frac{1}{9}\)) of the original.
• When the distance is quadrupled, as seen in our exercise, the force drops to one-sixteenth \(\left(\frac{1}{16}\right)\).

Understanding the way distance affects gravitational force allows scientists to predict celestial movements and the behavior of satellites in orbit.

mass and gravity

Mass plays a vital role in determining the strength of gravitational force. In the context of Newton's Universal Law of Gravitation, the mass of the objects directly influences the magnitude of the gravitational force. Larger masses generate stronger forces.
Consider these points:

• Doubling the mass of one of the objects will double the gravitational force. For example, if we replace the Sun with a star twice as massive, the gravitational force between it and Earth will also double, demonstrating a direct relationship between mass and gravity.
• If multiple masses are involved, the overall gravitational pull is the vector sum of individual gravitational forces exerted by each mass.

The interaction between mass and gravity is fundamental in areas such as planetary formation, orbit prediction, and even in theories related to black holes.