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Chapter 4: Problem 26

Suppose that the Earth were moved to a distance of \(3.0 \mathrm{AU}\) from the Sun. How much stronger or weaker would the Sun's gravitational pull be on the Earth? Explain.

Short Answer

Expert verified
The Sun's gravitational pull on Earth would be \(1/9\) times as strong if the Earth were moved to a distance of 3 AU from the Sun.

Step by step solution

01

Understand the Given Information

It is stated in the exercise that the Earth is moved to a distance of 3 AU from the Sun. 1 AU is the average distance from the Sun to the Earth, so 3 AU is three times the average Earth-Sun distance.
02

Apply Newton's Law of Universal Gravitation

According to Newton's Law of Universal Gravitation, the force between two objects is inversely proportional to the square of the distance between them. Therefore, if the distance increases by a factor of 3, the force decreases by a factor of \(3^2\) or 9.
03

Calculate the Change in Gravitational Force

Since the Earth is moved to 3 AU, the distance is three times greater. Therefore, the gravitational force is \(1/9\) as strong as it was at 1 AU.

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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 one of the fundamental forces in the universe. This force is the attraction between two masses. It is determined by Newton's Law of Universal Gravitation.
The formula for calculating gravitational force is:\[F = G \frac{m_1 m_2}{r^2}\]Here,
  • \(F\) is the gravitational force.
  • \(G\) is the gravitational constant, which is approximately \(6.674 \times 10^{-11} \text{Nm}^2/\text{kg}^2\).
  • \(m_1\) and \(m_2\) are the masses of the two objects.
  • \(r\) is the distance between the centers of the two masses.
Gravitational force acts equally on both objects and is crucial for keeping planets in orbit around stars, like the Earth orbiting the Sun.
Inverse Square Law
The inverse square law is a fundamental principle that applies to several physical phenomena, including gravity. It states that the strength of a force decreases with the square of the distance from the source.
In the case of gravitational force, as the distance \(r\) between two masses increases, the gravitational force \(F\) diminishes according to the formula.
  • When the distance doubles, the force reduces by a factor of four (i.e., \(2^2\)).
  • When the distance triples, the force reduces by a factor of nine (i.e., \(3^2\)).
For the Earth-Sun scenario mentioned in the exercise, by moving the Earth three times farther away, the force becomes nine times weaker. The understanding of this principle is pivotal in fields such as astronomy and physics.
Earth-Sun Distance
The average distance from Earth to the Sun is known as 1 Astronomical Unit (AU). This unit provides a convenient reference for measuring distances within our solar system. An AU is about 149.6 million kilometers (approximately 93 million miles).
This measurement is not only crucial for understanding our solar system's layout but also plays a role in calculating gravitational forces.
  • If the Earth was located at 3 AU from the Sun, the distance would be three times farther than usual.
  • This larger distance impacts the gravitational interaction, reducing the Sun's pull on Earth to a fraction (1/9) of its force at 1 AU distance, as derived from the inverse square law.
The concept of AU helps us comprehend vast celestial distances and the dynamic relationship between Earth and the Sun.

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Most popular questions from this chapter

What is parallax? What did Tycho Brahe conclude from his attempt to measure the parallax of a supernova and a comet?

The orbit of a spacecraft about the Sun has a perihelion distance of \(0.1 \mathrm{AU}\) and an aphelion distance of \(0.4 \mathrm{AU}\). What is the semimajor axis of the spacecraft's orbit? What is its orbital period?

A comet orbits the Sun with a sidereal period of \(64.0\) years. (a) Find the semimajor axis of the orbit. (b) At aphelion, the comet is \(31.5\) AU from the Sun. How far is it from the Sun at perihelion?

Use the Starry Night Enthusiast \({ }^{\mathrm{TM}}\) program to observe the changing appearance of Mercury. Display the entire celestial sphere (select Guides > Atlas in the Favourites menu) and center on Mercury (double-click the entry for Mercury in the Find pane); then use the zoom controls at the right-hand end of the toolbar (at the top of the main window) to adjust your view so that you can clearly see details on the planet's surface. (Click on the + button to zoom in and on the - button to zoom out.) (a) Click on the Time Flow Rate control (immediately to the right of the date and time display) and set the discrete time step to 1 day. Using the Step Forward button, observe and record the changes in Mercury's phase and apparent size from one day to the next. Run time forward for some time to see these changes more graphically. (b) Explain why the phase and apparent size change in the way that you observe.

Is it ever possible to see Mercury at midnight? Explain your answer.
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