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Chapter 4: Problem 42

The mass of Saturn is approximately 100 times that of Earth, and the semimajor axis of Saturn's orbit is approximately \(10 \mathrm{AU}\). To this approximation, how does the gravitational force that the Sun exerts on Saturn compare to the gravitational force that the Sun exerts on the Earth? How do the accelerations of Saturn and the Earth compare?

Short Answer

Expert verified
The Sun exerts the same gravitational force on Saturn as it does on Earth, however, the acceleration of Saturn is 1/100th that of Earth’s.

Step by step solution

01

Identify the Formula for Gravitational Force

The formula for gravitational force is given by Newton's law of gravitation: \(F = G \frac{{m1*m2}}{{r^2}}\) where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects and r is the distance between the centers of the two objects.
02

Calculate Force Ratio for Saturn and Earth

The ratio of the gravitational forces exerted on Saturn and Earth will be found by replacing m1 with the mass of Saturn, m2 with mass of the Sun, and r with the separation between Saturn and the Sun, and then doing the same for Earth. Given that the mass of Saturn is roughly 100 times that of the Earth and its distance from the Sun is about 10 times as great, the force ratio comes out to be \((100/10^2) = 1\). Therefore, the Sun exerts the same gravitational force on both Saturn and Earth.
03

Identify the Formula for Acceleration

Acceleration due to gravitational force is calculated by Newton's second law: \(a = F/m\), where a is the acceleration, F is the net force and m is the mass of the object.
04

Calculate Acceleration Ratio for Saturn and Earth

To compare the accelerations of Earth and Saturn, we use the formula for acceleration: \(a = F/m\). Here, F is the same for both Saturn and Earth, obtained from Step 2. From the given, we know that the mass of Saturn is 100 times that of Earth, hence, the acceleration of Saturn will be \(1/100\) times that of Earth.

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Most popular questions from this chapter

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.

Figure 4-21 shows the lunar module Eagle in orbit around the Moon after completing the first successful lunar landing in July 1969. (The photograph was taken from the command module Columbia, in which the astronauts returned to Earth.) The spacecraft orbited \(111 \mathrm{~km}\) above the surface of the Moon. Calculate the period of the spacecraft's orbit. See Appendix 3 for relevant data about the Moon.

What observations did Tycho Brahe make in an attempt to test the heliocentric model? What were his results? Explain why modern astronomers get different results.

What is the difference between weight and mass?

A certain comet is \(2 \mathrm{AU}\) from the Sun at perihelion and \(16 \mathrm{AU}\) from the Sun at aphelion. (a) Find the semimajor axis of the comet's orbit. (b) Find the sidereal period of the orbit.
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