CHAPTER 7

Newton's Law of Gravitation

Figure 7.1 Johannes Kepler (left) showed how the planets move, and Isaac Newton (right) discovered that
gravitational force caused them to move that way. ((left) unknown, Public Domain; (right) Sir Godfrey Kneller, Public
Domain)

Chapter Outline

7.1 Kepler's Laws of Planetary Motion

7.2 Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity

INTRODUCTION What do a falling apple and the orbit of the moon have in common? You will learn in this chapter that each is
caused by gravitational force. The motion of all celestial objects, in fact, is determined by the gravitational force, which depends
on their mass and separation.

Johannes Kepler discovered three laws of planetary motion that all orbiting planets and moons follow. Years later, Isaac Newton
found these laws useful in developing his law of universal gravitation. This law relates gravitational force to the masses of objects
and the distance between them. Many years later still, Albert Einstein showed there was a little more to the gravitation story
when he published his theory of general relativity.

7.1 Kepler's Laws of Planetary Motion

Section Learning Objectives
By the end of this section, you will be able to do the following:
• Explain Kepler’s three laws of planetary motion
• Apply Kepler’s laws to calculate characteristics of orbits
230 Chapter 7 • Newton's Law of Gravitation

Section Key Terms

aphelion Copernican model eccentricity

Kepler’s laws of planetary motion perihelion Ptolemaic model

Concepts Related to Kepler’s Laws of Planetary Motion

Examples of orbits abound. Hundreds of artificial satellites orbit Earth together with thousands of pieces of debris. The moon’s
orbit around Earth has intrigued humans from time immemorial. The orbits of planets, asteroids, meteors, and comets around
the sun are no less interesting. If we look farther, we see almost unimaginable numbers of stars, galaxies, and other celestial
objects orbiting one another and interacting through gravity.

All these motions are governed by gravitational force. The orbital motions of objects in our own solar system are simple enough
to describe with a few fairly simple laws. The orbits of planets and moons satisfy the following two conditions:

• The mass of the orbiting object, m, is small compared to the mass of the object it orbits, M.
• The system is isolated from other massive objects.

Based on the motion of the planets about the sun, Kepler devised a set of three classical laws, called Kepler’s laws of planetary
motion, that describe the orbits of all bodies satisfying these two conditions:

1. The orbit of each planet around the sun is an ellipse with the sun at one focus.
2. Each planet moves so that an imaginary line drawn from the sun to the planet sweeps out equal areas in equal times.
3. The ratio of the squares of the periods of any two planets about the sun is equal to the ratio of the cubes of their average
distances from the sun.

These descriptive laws are named for the German astronomer Johannes Kepler (1571–1630). He devised them after careful study
(over some 20 years) of a large amount of meticulously recorded observations of planetary motion done by Tycho Brahe
(1546–1601). Such careful collection and detailed recording of methods and data are hallmarks of good science. Data constitute
the evidence from which new interpretations and meanings can be constructed. Let’s look closer at each of these laws.

Kepler’s First Law

The orbit of each planet about the sun is an ellipse with the sun at one focus, as shown in Figure 7.2. The planet’s closest
approach to the sun is called aphelion and its farthest distance from the sun is called perihelion.

Access for free at openstax.org.

 7.1 • Kepler's Laws of Planetary Motion 231

Figure 7.2 (a) An ellipse is a closed curve such that the sum of the distances from a point on the curve to the two foci (f1 and f2) is constant.
(b) For any closed orbit, m follows an elliptical path with M at one focus. (c) The aphelion (ra) is the closest distance between the planet and
the sun, while the perihelion (rp) is the farthest distance from the sun.

If you know the aphelion (ra) and perihelion (rp) distances, then you can calculate the semi-major axis (a) and semi-minor axis
(b).

Figure 7.3 You can draw an ellipse as shown by putting a pin at each focus, and then placing a loop of string around a pen and the pins and
tracing a line on the paper.
232 Chapter 7 • Newton's Law of Gravitation

Kepler’s Second Law

Each planet moves so that an imaginary line drawn from the sun to the planet sweeps out equal areas in equal times, as shown
in Figure 7.4.

Figure 7.4 The shaded regions have equal areas. The time for m to go from A to B is the same as the time to go from C to D and from E to F.
The mass m moves fastest when it is closest to M. Kepler’s second law was originally devised for planets orbiting the sun, but it has broader
validity.

TIPS FOR SUCCESS

Note that while, for historical reasons, Kepler’s laws are stated for planets orbiting the sun, they are actually valid for all
bodies satisfying the two previously stated conditions.

Kepler’s Third Law

The ratio of the periods squared of any two planets around the sun is equal to the ratio of their average distances from the sun
cubed. In equation form, this is

where T is the period (time for one orbit) and r is the average distance (also called orbital radius). This equation is valid only for
comparing two small masses orbiting a single large mass. Most importantly, this is only a descriptive equation; it gives no
information about the cause of the equality.

LINKS TO PHYSICS

History: Ptolemy vs. Copernicus

Before the discoveries of Kepler, Copernicus, Galileo, Newton, and others, the solar system was thought to revolve around Earth
as shown in Figure 7.5 (a). This is called the Ptolemaic model, named for the Greek philosopher Ptolemy who lived in the second
century AD. The Ptolemaic model is characterized by a list of facts for the motions of planets, with no explanation of cause and
effect. There tended to be a different rule for each heavenly body and a general lack of simplicity.

Figure 7.5 (b) represents the modern or Copernican model. In this model, a small set of rules and a single underlying force
explain not only all planetary motion in the solar system, but also all other situations involving gravity. The breadth and
simplicity of the laws of physics are compelling.

Access for free at openstax.org.

 7.1 • Kepler's Laws of Planetary Motion 233

Figure 7.5 (a) The Ptolemaic model of the universe has Earth at the center with the moon, the planets, the sun, and the stars revolving
about it in complex circular paths. This geocentric (Earth-centered) model, which can be made progressively more accurate by adding more
circles, is purely descriptive, containing no hints about the causes of these motions. (b) The Copernican heliocentric (sun-centered) model
is a simpler and more accurate model.

Nicolaus Copernicus (1473–1543) first had the idea that the planets circle the sun, in about 1514. It took him almost 20 years to
work out the mathematical details for his model. He waited another 10 years or so to publish his work. It is thought he hesitated
because he was afraid people would make fun of his theory. Actually, the reaction of many people was more one of fear and
anger. Many people felt the Copernican model threatened their basic belief system. About 100 years later, the astronomer Galileo
was put under house arrest for providing evidence that planets, including Earth, orbited the sun. In all, it took almost 300 years
for everyone to admit that Copernicus had been right all along.

GRASP CHECK
Explain why Earth does actually appear to be the center of the solar system.
a. Earth appears to be the center of the solar system because Earth is at the center of the universe, and everything revolves
around it in a circular orbit.
b. Earth appears to be the center of the solar system because, in the reference frame of Earth, the sun, moon, and planets
all appear to move across the sky as if they were circling Earth.
c. Earth appears to be at the center of the solar system because Earth is at the center of the solar system and all the
heavenly bodies revolve around it.
d. Earth appears to be at the center of the solar system because Earth is located at one of the foci of the elliptical orbit of
the sun, moon, and other planets.

Virtual Physics
Acceleration
This simulation allows you to create your own solar system so that you can see how changing distances and masses
determines the orbits of planets. Click Help for instructions.

Click to view content (https://archive.cnx.org/specials/ee816dff-0b5f-4e6f-8250-f9fb9e39d716/my-solar-system/)

GRASP CHECK
When the central object is off center, how does the speed of the orbiting object vary?
a. The orbiting object moves fastest when it is closest to the central object and slowest when it is farthest away.
b. The orbiting object moves slowest when it is closest to the central object and fastest when it is farthest away.
234 Chapter 7 • Newton's Law of Gravitation

c. The orbiting object moves with the same speed at every point on the circumference of the elliptical orbit.
d. There is no relationship between the speed of the object and the location of the planet on the circumference of the
orbit.

Calculations Related to Kepler’s Laws of Planetary Motion

Kepler’s First Law
Refer back to Figure 7.2 (a). Notice which distances are constant. The foci are fixed, so distance is a constant. The definition
of an ellipse states that the sum of the distances is also constant. These two facts taken together mean that the
perimeter of triangle must also be constant. Knowledge of these constants will help you determine positions and
distances of objects in a system that includes one object orbiting another.

Kepler’s Second Law

Refer back to Figure 7.4. The second law says that the segments have equal area and that it takes equal time to sweep through
each segment. That is, the time it takes to travel from A to B equals the time it takes to travel from C to D, and so forth. Velocity v
equals distance d divided by time t: . Then, , so distance divided by velocity is also a constant. For example, if we
know the average velocity of Earth on June 21 and December 21, we can compare the distance Earth travels on those days.

The degree of elongation of an elliptical orbit is called its eccentricity (e). Eccentricity is calculated by dividing the distance f
from the center of an ellipse to one of the foci by half the long axis a.
7.1
When , the ellipse is a circle.

The area of an ellipse is given by , where b is half the short axis. If you know the axes of Earth’s orbit and the area Earth
sweeps out in a given period of time, you can calculate the fraction of the year that has elapsed.

WORKED EXAMPLE

Kepler’s First Law

At its closest approach, a moon comes within 200,000 km of the planet it orbits. At that point, the moon is 300,000 km from the
other focus of its orbit, f2. The planet is focus f1 of the moon’s elliptical orbit. How far is the moon from the planet when it is
260,000 km from f2?
Strategy
Show and label the ellipse that is the orbit in your solution. Picture the triangle f1mf2 collapsed along the major axis and add up
the lengths of the three sides. Find the length of the unknown side of the triangle when the moon is 260,000 km from f2.

Solution
Perimeter of

Discussion
The perimeter of triangle f1mf2 must be constant because the distance between the foci does not change and Kepler’s first law
says the orbit is an ellipse. For any ellipse, the sum of the two sides of the triangle, which are f1m and mf2, is constant.

WORKED EXAMPLE

Kepler’s Second Law

Figure 7.6 shows the major and minor axes of an ellipse. The semi-major and semi-minor axes are half of these, respectively.

Access for free at openstax.org.

 7.1 • Kepler's Laws of Planetary Motion 235

Figure 7.6 The major axis is the length of the ellipse, and the minor axis is the width of the ellipse. The semi-major axis is half the major
axis, and the semi-minor axis is half the minor axis.

Earth’s orbit is slightly elliptical, with a semi-major axis of 152 million km and a semi-minor axis of 147 million km. If Earth’s
period is 365.26 days, what area does an Earth-to-sun line sweep past in one day?
Strategy
Each day, Earth sweeps past an equal-sized area, so we divide the total area by the number of days in a year to find the area
swept past in one day. For total area use . Calculate A, the area inside Earth’s orbit and divide by the number of days in
a year (i.e., its period).

Solution

7.2

The area swept out in one day is thus .

Discussion
The answer is based on Kepler’s law, which states that a line from a planet to the sun sweeps out equal areas in equal times.

Kepler’s Third Law

Kepler’s third law states that the ratio of the squares of the periods of any two planets (T1, T2) is equal to the ratio of the cubes of
their average orbital distance from the sun (r1, r2). Mathematically, this is represented by

From this equation, it follows that the ratio r3/T2 is the same for all planets in the solar system. Later we will see how the work of
Newton leads to a value for this constant.

WORKED EXAMPLE

Kepler’s Third Law

Given that the moon orbits Earth each 27.3 days and that it is an average distance of from the center of Earth,
calculate the period of an artificial satellite orbiting at an average altitude of 1,500 km above Earth’s surface.
Strategy
The period, or time for one orbit, is related to the radius of the orbit by Kepler’s third law, given in mathematical form by
236 Chapter 7 • Newton's Law of Gravitation

. Let us use the subscript 1 for the moon and the subscript 2 for the satellite. We are asked to find T2. The given
information tells us that the orbital radius of the moon is , and that the period of the moon is
. The height of the artificial satellite above Earth’s surface is given, so to get the distance r2 from the center of
Earth we must add the height to the radius of Earth (6380 km). This gives . Now all
quantities are known, so T2 can be found.

Solution
To solve for T2, we cross-multiply and take the square root, yielding

7.3

Discussion
This is a reasonable period for a satellite in a fairly low orbit. It is interesting that any satellite at this altitude will complete one
orbit in the same amount of time.

Practice Problems
1. A planet with no axial tilt is located in another solar system. It circles its sun in a very elliptical orbit so that the temperature
varies greatly throughout the year. If the year there has 612 days and the inhabitants celebrate the coldest day on day 1 of their
calendar, when is the warmest day?
a. Day 1
b. Day 153
c. Day 306
d. Day 459

2. A geosynchronous Earth satellite is one that has an orbital period of precisely 1 day. Such orbits are useful for
communication and weather observation because the satellite remains above the same point on Earth (provided it orbits in
the equatorial plane in the same direction as Earth’s rotation). The ratio for the moon is . Calculate the
radius of the orbit of such a satellite.
a.
b.
c.
d.

Check Your Understanding

3. Are Kepler’s laws purely descriptive, or do they contain causal information?
a. Kepler’s laws are purely descriptive.
b. Kepler’s laws are purely causal.
c. Kepler’s laws are descriptive as well as causal.
d. Kepler’s laws are neither descriptive nor causal.

4. True or false—According to Kepler’s laws of planetary motion, a satellite increases its speed as it approaches its parent body
and decreases its speed as it moves away from the parent body.
a. True
b. False

5. Identify the locations of the foci of an elliptical orbit.

a. One focus is the parent body, and the other is located at the opposite end of the ellipse, at the same distance from the
center as the parent body.
b. One focus is the parent body, and the other is located at the opposite end of the ellipse, at half the distance from the
center as the parent body.

Access for free at openstax.org.

 7.2 • Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity 237

c. One focus is the parent body and the other is located outside of the elliptical orbit, on the line on which is the semi-
major axis of the ellipse.
d. One focus is on the line containing the semi-major axis of the ellipse, and the other is located anywhere on the elliptical
orbit of the satellite.

7.2 Newton's Law of Universal Gravitation and Einstein's

Theory of General Relativity
Section Learning Objectives
By the end of this section, you will be able to do the following:
• Explain Newton’s law of universal gravitation and compare it to Einstein’s theory of general relativity
• Perform calculations using Newton’s law of universal gravitation

Section Key Terms

Einstein’s theory of general relativity gravitational constant Newton’s universal law of gravitation

Concepts Related to Newton’s Law of Universal Gravitation

Sir Isaac Newton was the first scientist to precisely define the gravitational force, and to show that it could explain both falling
bodies and astronomical motions. See Figure 7.7. But Newton was not the first to suspect that the same force caused both our
weight and the motion of planets. His forerunner, Galileo Galilei, had contended that falling bodies and planetary motions had
the same cause. Some of Newton’s contemporaries, such as Robert Hooke, Christopher Wren, and Edmund Halley, had also
made some progress toward understanding gravitation. But Newton was the first to propose an exact mathematical form and to
use that form to show that the motion of heavenly bodies should be conic sections—circles, ellipses, parabolas, and hyperbolas.
This theoretical prediction was a major triumph. It had been known for some time that moons, planets, and comets follow such
paths, but no one had been able to propose an explanation of the mechanism that caused them to follow these paths and not
others.

Figure 7.7 The popular legend that Newton suddenly discovered the law of universal gravitation when an apple fell from a tree and hit him
on the head has an element of truth in it. A more probable account is that he was walking through an orchard and wondered why all the
apples fell in the same direction with the same acceleration. Great importance is attached to it because Newton’s universal law of
gravitation and his laws of motion answered very old questions about nature and gave tremendous support to the notion of underlying
simplicity and unity in nature. Scientists still expect underlying simplicity to emerge from their ongoing inquiries into nature.

The gravitational force is relatively simple. It is always attractive, and it depends only on the masses involved and the distance
238 Chapter 7 • Newton's Law of Gravitation

between them. Expressed in modern language, Newton’s universal law of gravitation states that every object in the universe
attracts every other object with a force that is directed along a line joining them. The force is directly proportional to the product
of their masses and inversely proportional to the square of the distance between them. This attraction is illustrated by Figure
7.8.

Figure 7.8 Gravitational attraction is along a line joining the centers of mass (CM) of the two bodies. The magnitude of the force on each
body is the same, consistent with Newton’s third law (action-reaction).

For two bodies having masses m and M with a distance r between their centers of mass, the equation for Newton’s universal law
of gravitation is

where F is the magnitude of the gravitational force and G is a proportionality factor called the gravitational constant. G is a
universal constant, meaning that it is thought to be the same everywhere in the universe. It has been measured experimentally
to be .

If a person has a mass of 60.0 kg, what would be the force of gravitational attraction on him at Earth’s surface? G is given above,
Earth’s mass M is 5.97 × 1024 kg, and the radius r of Earth is 6.38 × 106 m. Putting these values into Newton’s universal law of
gravitation gives

We can check this result with the relationship:

You may remember that g, the acceleration due to gravity, is another important constant related to gravity. By substituting g for
a in the equation for Newton’s second law of motion we get . Combining this with the equation for universal
gravitation gives

Cancelling the mass m on both sides of the equation and filling in the values for the gravitational constant and mass and radius
of the Earth, gives the value of g, which may look familiar.

This is a good point to recall the difference between mass and weight. Mass is the amount of matter in an object; weight is the

Access for free at openstax.org.

 7.2 • Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity 239

force of attraction between the mass within two objects. Weight can change because g is different on every moon and planet. An
object’s mass m does not change but its weight mg can.

Virtual Physics
Gravity and Orbits
Move the sun, Earth, moon and space station in this simulation to see how it affects their gravitational forces and orbital
paths. Visualize the sizes and distances between different heavenly bodies. Turn off gravity to see what would happen
without it!

Click to view content (https://archive.cnx.org/specials/a14085c8-96b8-4d04-bb5a-56d9ccbe6e69/gravity-and-orbits/)

GRASP CHECK
Why doesn’t the Moon travel in a smooth circle around the Sun?
a. The Moon is not affected by the gravitational field of the Sun.
b. The Moon is not affected by the gravitational field of the Earth.
c. The Moon is affected by the gravitational fields of both the Earth and the Sun, which are always additive.
d. The moon is affected by the gravitational fields of both the Earth and the Sun, which are sometimes additive and
sometimes opposite.

Snap Lab
Take-Home Experiment: Falling Objects
In this activity you will study the effects of mass and air resistance on the acceleration of falling objects. Make predictions
(hypotheses) about the outcome of this experiment. Write them down to compare later with results.

• Four sheets of -inch paper

Procedure
• Take four identical pieces of paper.
◦ Crumple one up into a small ball.
◦ Leave one uncrumpled.
◦ Take the other two and crumple them up together, so that they make a ball of exactly twice the mass of the other
crumpled ball.
◦ Now compare which ball of paper lands first when dropped simultaneously from the same height.
1. Compare crumpled one-paper ball with crumpled two-paper ball.
2. Compare crumpled one-paper ball with uncrumpled paper.

GRASP CHECK
Why do some objects fall faster than others near the surface of the earth if all mass is attracted equally by the force of
gravity?
a. Some objects fall faster because of air resistance, which acts in the direction of the motion of the object and exerts
more force on objects with less surface area.
b. Some objects fall faster because of air resistance, which acts in the direction opposite the motion of the object and
exerts more force on objects with less surface area.
c. Some objects fall faster because of air resistance, which acts in the direction of motion of the object and exerts more
force on objects with more surface area.
d. Some objects fall faster because of air resistance, which acts in the direction opposite the motion of the object and
exerts more force on objects with more surface area.

It is possible to derive Kepler’s third law from Newton’s law of universal gravitation. Applying Newton’s second law of motion to
240 Chapter 7 • Newton's Law of Gravitation

angular motion gives an expression for centripetal force, which can be equated to the expression for force in the universal
gravitation equation. This expression can be manipulated to produce the equation for Kepler’s third law. We saw earlier that the
expression r3/T2 is a constant for satellites orbiting the same massive object. The derivation of Kepler’s third law from Newton’s
law of universal gravitation and Newton’s second law of motion yields that constant:

where M is the mass of the central body about which the satellites orbit (for example, the sun in our solar system). The
usefulness of this equation will be seen later.

The universal gravitational constant G is determined experimentally. This definition was first done accurately in 1798 by English
scientist Henry Cavendish (1731–1810), more than 100 years after Newton published his universal law of gravitation. The
measurement of G is very basic and important because it determines the strength of one of the four forces in nature.
Cavendish’s experiment was very difficult because he measured the tiny gravitational attraction between two ordinary-sized
masses (tens of kilograms at most) by using an apparatus like that in Figure 7.9. Remarkably, his value for G differs by less than
1% from the modern value.

Figure 7.9 Cavendish used an apparatus like this to measure the gravitational attraction between two suspended spheres (m) and two
spheres on a stand (M) by observing the amount of torsion (twisting) created in the fiber. The distance between the masses can be varied to
check the dependence of the force on distance. Modern experiments of this type continue to explore gravity.

Einstein’s Theory of General Relativity

Einstein’s theory of general relativity explained some interesting properties of gravity not covered by Newton’s theory. Einstein
based his theory on the postulate that acceleration and gravity have the same effect and cannot be distinguished from each
other. He concluded that light must fall in both a gravitational field and in an accelerating reference frame. Figure 7.10 shows
this effect (greatly exaggerated) in an accelerating elevator. In Figure 7.10(a), the elevator accelerates upward in zero gravity. In
Figure 7.10(b), the room is not accelerating but is subject to gravity. The effect on light is the same: it “falls” downward in both
situations. The person in the elevator cannot tell whether the elevator is accelerating in zero gravity or is stationary and subject
to gravity. Thus, gravity affects the path of light, even though we think of gravity as acting between masses, while photons are
massless.

Access for free at openstax.org.

 7.2 • Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity 241

Figure 7.10 (a) A beam of light emerges from a flashlight in an upward-accelerating elevator. Since the elevator moves up during the time
the light takes to reach the wall, the beam strikes lower than it would if the elevator were not accelerated. (b) Gravity must have the same
effect on light, since it is not possible to tell whether the elevator is accelerating upward or is stationary and acted upon by gravity.

Einstein’s theory of general relativity got its first verification in 1919 when starlight passing near the sun was observed during a
solar eclipse. (See Figure 7.11.) During an eclipse, the sky is darkened and we can briefly see stars. Those on a line of sight nearest
the sun should have a shift in their apparent positions. Not only was this shift observed, but it agreed with Einstein’s predictions
well within experimental uncertainties. This discovery created a scientific and public sensation. Einstein was now a folk hero as
well as a very great scientist. The bending of light by matter is equivalent to a bending of space itself, with light following the
curve. This is another radical change in our concept of space and time. It is also another connection that any particle with mass
or energy (e.g., massless photons) is affected by gravity.

Figure 7.11 This schematic shows how light passing near a massive body like the sun is curved toward it. The light that reaches the Earth
then seems to be coming from different locations than the known positions of the originating stars. Not only was this effect observed, but
the amount of bending was precisely what Einstein predicted in his general theory of relativity.

To summarize the two views of gravity, Newton envisioned gravity as a tug of war along the line connecting any two objects in
the universe. In contrast, Einstein envisioned gravity as a bending of space-time by mass.
242 Chapter 7 • Newton's Law of Gravitation

BOUNDLESS PHYSICS

NASA gravity probe B

NASA’s Gravity Probe B (GP-B) mission has confirmed two key predictions derived from Albert Einstein’s general theory of
relativity. The probe, shown in Figure 7.12 was launched in 2004. It carried four ultra-precise gyroscopes designed to measure
two effects hypothesized by Einstein’s theory:

• The geodetic effect, which is the warping of space and time by the gravitational field of a massive body (in this case, Earth)
• The frame-dragging effect, which is the amount by which a spinning object pulls space and time with it as it rotates

Figure 7.12 Artist concept of Gravity Probe B spacecraft in orbit around the Earth. (credit: NASA/MSFC)

Both effects were measured with unprecedented precision. This was done by pointing the gyroscopes at a single star while
orbiting Earth in a polar orbit. As predicted by relativity theory, the gyroscopes experienced very small, but measureable,
changes in the direction of their spin caused by the pull of Earth’s gravity.

The principle investigator suggested imagining Earth spinning in honey. As Earth rotates it drags space and time with it as it
would a surrounding sea of honey.

GRASP CHECK
According to the general theory of relativity, a gravitational field bends light. What does this have to do with time and space?
a. Gravity has no effect on the space-time continuum, and gravity only affects the motion of light.
b. The space-time continuum is distorted by gravity, and gravity has no effect on the motion of light.
c. Gravity has no effect on either the space-time continuum or on the motion of light.
d. The space-time continuum is distorted by gravity, and gravity affects the motion of light.

Calculations Based on Newton’s Law of Universal Gravitation

TIPS FOR SUCCESS
When performing calculations using the equations in this chapter, use units of kilograms for mass, meters for distances,
newtons for force, and seconds for time.

The mass of an object is constant, but its weight varies with the strength of the gravitational field. This means the value of g
varies from place to place in the universe. The relationship between force, mass, and acceleration from the second law of motion
can be written in terms of g.

In this case, the force is the weight of the object, which is caused by the gravitational attraction of the planet or moon on which
the object is located. We can use this expression to compare weights of an object on different moons and planets.

Access for free at openstax.org.

 7.2 • Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity 243

WATCH PHYSICS

Mass and Weight Clarification

This video shows the mathematical basis of the relationship between mass and weight. The distinction between mass and weight
are clearly explained. The mathematical relationship between mass and weight are shown mathematically in terms of the
equation for Newton’s law of universal gravitation and in terms of his second law of motion.

Click to view content (https://www.khanacademy.org/embed_video?v=IuBoeDihLUc)

GRASP CHECK
Would you have the same mass on the moon as you do on Earth? Would you have the same weight?
a. You would weigh more on the moon than on Earth because gravity on the moon is stronger than gravity on Earth.
b. You would weigh less on the moon than on Earth because gravity on the moon is weaker than gravity on Earth.
c. You would weigh less on the moon than on Earth because gravity on the moon is stronger than gravity on Earth.
d. You would weigh more on the moon than on Earth because gravity on the moon is weaker than gravity on Earth.

Two equations involving the gravitational constant, G, are often useful. The first is Newton’s equation, . Several of
the values in this equation are either constants or easily obtainable. F is often the weight of an object on the surface of a large
object with mass M, which is usually known. The mass of the smaller object, m, is often known, and G is a universal constant
with the same value anywhere in the universe. This equation can be used to solve problems involving an object on or orbiting
Earth or other massive celestial object. Sometimes it is helpful to equate the right-hand side of the equation to mg and cancel
the m on both sides.

The equation is also useful for problems involving objects in orbit. Note that there is no need to know the mass of the
object. Often, we know the radius r or the period T and want to find the other. If these are both known, we can use the equation
to calculate the mass of a planet or star.

WATCH PHYSICS

Mass and Weight Clarification

This video demonstrates calculations involving Newton’s universal law of gravitation.

Click to view content (https://www.khanacademy.org/embed_video?v=391txUI76gM)

GRASP CHECK
Identify the constants and .
a. and are both the acceleration due to gravity
b. is acceleration due to gravity on Earth and is the universal gravitational constant.
c. is the gravitational constant and is the acceleration due to gravity on Earth.
d. and are both the universal gravitational constant.

WORKED EXAMPLE

Change in g
The value of g on the planet Mars is 3.71 m/s2. If you have a mass of 60.0 kg on Earth, what would be your mass on Mars? What
would be your weight on Mars?
Strategy
Weight equals acceleration due to gravity times mass: . An object’s mass is constant. Call acceleration due to gravity
on Mars gM and weight on Mars WM.
244 Chapter 7 • Newton's Law of Gravitation

Solution
Mass on Mars would be the same, 60 kg.

7.4

Discussion
The value of g on any planet depends on the mass of the planet and the distance from its center. If the material below the surface
varies from point to point, the value of g will also vary slightly.

WORKED EXAMPLE

Earth’s g at the Moon

Find the acceleration due to Earth’s gravity at the distance of the moon.

7.5

Express the force of gravity in terms of g.

7.6
Combine with the equation for universal gravitation.
7.7

Solution
Cancel m and substitute.

7.8

Discussion
The value of g for the moon is 1.62 m/s2. Comparing this value to the answer, we see that Earth’s gravitational influence on an
object on the moon’s surface would be insignificant.

Practice Problems
6. What is the mass of a person who weighs ?
a.
b.
c.
d.

7. Calculate Earth’s mass given that the acceleration due to gravity at the North Pole is and the radius of the Earth is
from pole to center.
a.
b.
c.
d.

Check Your Understanding

8. Some of Newton’s predecessors and contemporaries also studied gravity and proposed theories. What important advance
did Newton make in the study of gravity that the other scientists had failed to do?
a. He gave an exact mathematical form for the theory.

Access for free at openstax.org.

 7.2 • Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity 245

b. He added a correction term to a previously existing formula.

c. Newton found the value of the universal gravitational constant.
d. Newton showed that gravitational force is always attractive.

9. State the law of universal gravitation in words only.

a. Gravitational force between two objects is directly proportional to the sum of the squares of their masses and inversely
proportional to the square of the distance between them.
b. Gravitational force between two objects is directly proportional to the product of their masses and inversely
proportional to the square of the distance between them.
c. Gravitational force between two objects is directly proportional to the sum of the squares of their masses and inversely
proportional to the distance between them.
d. Gravitational force between two objects is directly proportional to the product of their masses and inversely
proportional to the distance between them.

10. Newton’s law of universal gravitation explains the paths of what?

a. A charged particle
b. A ball rolling on a plane surface
c. A planet moving around the sun
d. A stone tied to a string and whirled at constant speed in a horizontal circle
246 Chapter 7 • Key Terms

KEY TERMS
aphelion closest distance between a planet and the sun Johannes Kepler that describe the properties of all
(called apoapsis for other celestial bodies) orbiting satellites
Copernican model the model of the solar system where the Newton’s universal law of gravitation states that
sun is at the center of the solar system and all the planets gravitational force between two objects is directly
orbit around it; this is also called the heliocentric model proportional to the product of their masses and inversely
eccentricity a measure of the separation of the foci of an proportional to the square of the distance between them.
ellipse perihelion farthest distance between a planet and the sun
Einstein’s theory of general relativity the theory that (called periapsis for other celestial bodies)
gravitational force results from the bending of spacetime Ptolemaic model the model of the solar system where Earth
by an object’s mass is at the center of the solar system and the sun and all the
gravitational constant the proportionality constant in planets orbit around it; this is also called the geocentric
Newton’s law of universal gravitation model
Kepler’s laws of planetary motion three laws derived by

SECTION SUMMARY
7.1 Kepler's Laws of Planetary 7.2 Newton's Law of Universal
Motion Gravitation and Einstein's Theory of
• All satellites follow elliptical orbits. General Relativity
• The line from the satellite to the parent body sweeps out • Newton’s law of universal gravitation provides a
equal areas in equal time. mathematical basis for gravitational force and Kepler’s
• The radius cubed divided by the period squared is a laws of planetary motion.
constant for all satellites orbiting the same parent body. • Einstein’s theory of general relativity shows that
gravitational fields change the path of light and warp
space and time.
• An object’s mass is constant, but its weight changes
when acceleration due to gravity, g, changes.

KEY EQUATIONS
7.1 Kepler's Laws of Planetary 7.2 Newton's Law of Universal
Motion Gravitation and Einstein's Theory of
General Relativity
Kepler’s third law
Newton’s second law of motion

eccentricity
Newton’s universal law of gravitation

area of an ellipse
acceleration due to gravity

semi-major axis of an ellipse

constant for satellites orbiting the
same massive object
semi-minor axis of an ellipse

CHAPTER REVIEW
Concept Items is different from other ellipses.
a. The foci of a circle are at the same point and are
7.1 Kepler's Laws of Planetary Motion located at the center of the circle.
1. A circle is a special case of an ellipse. Explain how a circle b. The foci of a circle are at the same point and are

Access for free at openstax.org.

 Chapter 7 • Chapter Review 247

located at the circumference of the circle. illustration of this is any description of the feeling of
c. The foci of a circle are at the same point and are constant velocity in a situation where no outside
located outside of the circle. frame of reference is considered.
d. The foci of a circle are at the same point and are c. Gravity and acceleration have the same effect and
located anywhere on the diameter, except on its cannot be distinguished from each other. An
midpoint. acceptable illustration of this is any description of
the feeling of acceleration in a situation where no
2. Comets have very elongated elliptical orbits with the sun
outside frame of reference is considered.
at one focus. Using Kepler's Law, explain why a comet
d. Gravity and acceleration have different effects and
travels much faster near the sun than it does at the other
can be distinguished from each other. An acceptable
end of the orbit.
illustration of this is any description of the feeling of
a. Because the satellite sweeps out equal areas in equal
acceleration in a situation where no outside frame of
times
reference is considered.
b. Because the satellite sweeps out unequal areas in
equal times 6. Titan, with a radius of , is the largest
c. Because the satellite is at the other focus of the moon of the planet Saturn. If the mass of Titan is
ellipse , what is the acceleration due to gravity
d. Because the square of the period of the satellite is on the surface of this moon?
proportional to the cube of its average distance from a.
the sun b.
3. True or False—A planet-satellite system must be isolated c.
from other massive objects to follow Kepler’s laws of d.
planetary motion. 7. Saturn’s moon Titan has an orbital period of 15.9 days. If
a. True Saturn has a mass of 5.68×1023 kg, what is the average
b. False distance from Titan to the center of Saturn?
4. Explain why the string, pins, and pencil method works a. 1.22×106 m
for drawing an ellipse. b. 4.26×107 m
a. The string, pins, and pencil method works because c. 5.25×104 km
the length of the two sides of the triangle remains d. 4.26×1010 km
constant as you are drawing the ellipse. 8. Explain why doubling the mass of an object doubles its
b. The string, pins, and pencil method works because weight, but doubling its distance from the center of
the area of the triangle remains constant as you are Earth reduces its weight fourfold.
drawing the ellipse. a. The weight is two times the gravitational force
c. The string, pins, and pencil method works because between the object and Earth.
the perimeter of the triangle remains constant as b. The weight is half the gravitational force between
you are drawing the ellipse. the object and Earth.
d. The string, pins, and pencil method works because c. The weight is equal to the gravitational force
the volume of the triangle remains constant as you between the object and Earth, and the gravitational
are drawing the ellipse. force is inversely proportional to the distance
squared between the object and Earth.
7.2 Newton's Law of Universal Gravitation d. The weight is directly proportional to the square of
and Einstein's Theory of General Relativity the gravitational force between the object and Earth.
5. Describe the postulate on which Einstein based the 9. Explain why a star on the other side of the Sun might
theory of general relativity and describe an everyday appear to be in a location that is not its true location.
experience that illustrates this postulate. a. It can be explained by using the concept of
a. Gravity and velocity have the same effect and cannot atmospheric refraction.
be distinguished from each other. An acceptable b. It can be explained by using the concept of the
illustration of this is any description of the feeling of special theory of relativity.
constant velocity in a situation where no outside c. It can be explained by using the concept of the
frame of reference is considered. general theory of relativity.
b. Gravity and velocity have different effects and can be d. It can be explained by using the concept of light
distinguished from each other. An acceptable
248 Chapter 7 • Chapter Review

scattering in the atmosphere. constant, G. Part B. Gravity is a very weak force

but, despite this limitation, Cavendish was able to
10. The Cavendish experiment marked a milestone in the
measure the attraction between very massive
study of gravity.
objects.
Part A. What important value did the experiment
c. Part A. The experiment measured the acceleration
determine?
due to gravity, g. Part B. Gravity is a very weak force
Part B. Why was this so difficult in terms of the masses
but despite this limitation, Cavendish was able to
used in the apparatus and the strength of the
measure the attraction between less massive
gravitational force?
objects.
a. Part A. The experiment measured the acceleration
d. Part A. The experiment measured the gravitational
due to gravity, g. Part B. Gravity is a very weak force
constant, G. Part B. Gravity is a very weak force but
but despite this limitation, Cavendish was able to
despite this limitation, Cavendish was able to
measure the attraction between very massive
measure the attraction between less massive
objects.
objects.
b. Part A. The experiment measured the gravitational

Critical Thinking Items

7.1 Kepler's Laws of Planetary Motion
11. In the figure, the time it takes for the planet to go from A
to B, C to D, and E to F is the same.

a. Area X < Area Y; the speed is greater for area X.

b. Area X > Area Y; the speed is greater for area Y.
c. Area X = Area Y; the speed is greater for area X.
d. Area X = Area Y; the speed is greater for area Y.

7.2 Newton's Law of Universal Gravitation

and Einstein's Theory of General Relativity
Compare the areas A1, A2, and A3 in terms of size.
14. Rhea, with a radius of 7.63×105 m, is the second-largest
a. A1 ≠ A2 ≠ A3
moon of the planet Saturn. If the mass of Rhea is
b. A1 = A2 = A3
2.31×1021 kg, what is the acceleration due to gravity on
c. A1 = A2 > A3
the surface of this moon?
d. A1 > A2 = A3
a. 2.65×10−1 m/s
12. A moon orbits a planet in an elliptical orbit. The foci of b. 2.02×105 m/s
the ellipse are 50, 000 km apart. The closest approach of c. 2.65×10−1 m/s2
the moon to the planet is 400, 000 km. What is the d. 2.02×105 m/s2
length of the major axis of the orbit?
15. Earth has a mass of 5.971×1024 kg and a radius of
a. 400, 000 km
6.371×106 m. Use the data to check the value of the
b. 450, 000, km
gravitational constant.
c. 800, 000 km
a. it matches the value of the
d. 850, 000 km
gravitational constant G.
13. In this figure, if f1 represents the parent body, which set b. it matches the value of the
of statements holds true?
gravitational constant G.
c. it matches the value of the
gravitational constant G.

Access for free at openstax.org.

 Chapter 7 • Test Prep 249

d. it matches the value of the a. 3.43×1019 kg

b. 1.99×1030 kg
gravitational constant G.
c. 2.56×1029 kg
16. The orbit of the planet Mercury has a period of 88.0 days d. 1.48×1040 kg
and an average radius of 5.791×1010 m. What is the mass
of the sun?

Problems 18. Earth is 1.496×108 km from the sun, and Neptune is

4.490×109 km from the sun. What best represents the
7.1 Kepler's Laws of Planetary Motion number of Earth years it takes for Neptune to complete
17. The closest Earth comes to the sun is 1.47×108 km, and one orbit around the sun?
Earth’s farthest distance from the sun is 1.52×108 km. a. 10 years
What is the area inside Earth’s orbit? b. 30 years
a. 2.23×1016 km2 c. 160 years
b. 6.79×1016 km2 d. 900 years
c. 7.02×1016 km2
d. 7.26×1016 km2

Performance Task • Two strong, permanent bar magnets

• A spring scale that can measure small forces
7.2 Newton's Law of Universal Gravitation • A short ruler calibrated in millimeters
and Einstein's Theory of General Relativity
Use the magnets to study the relationship between
19. Design an experiment to test whether magnetic force is attractive force and distance.
inversely proportional to the square of distance. a. What will be the independent variable?
Gravitational, magnetic, and electrical fields all act at a b. What will be the dependent variable?
distance, but do they all follow the inverse square law? c. How will you measure each of these variables?
One difference in the forces related to these fields is that d. If you plot the independent variable versus the
gravity is only attractive, but the other two can repel as dependent variable and the inverse square law is
well. In general, the inverse square law says that force F upheld, will the plot be a straight line? Explain.
equals a constant C divided by the distance between e. Which plot would be a straight line if the inverse
objects, d, squared: . square law were upheld?
Incorporate these materials into your design:

TEST PREP
Multiple Choice 22. An artificial satellite orbits the Earth at a distance of
1.45×104 km from Earth’s center. The moon orbits the
7.1 Kepler's Laws of Planetary Motion Earth at a distance of 3.84×105 km once every 27.3 days.
20. A planet of mass m circles a sun of mass M. Which How long does it take the satellite to orbit the Earth?
distance changes throughout the planet’s orbit? a. 0.200 days
a. b. 3.07 days
b. c. 243 days
c. d. 3721 days
d. 23. Earth is 1.496×108 km from the sun, and Venus is
21. The focal point of the elliptical orbit of a moon is 1.08×108 km from the sun. One day on Venus is 243
from the center of the orbit. If the Earth days long. What best represents the number of
eccentricity of the orbit is , what is the length of the Venusian days in a Venusian year?
semi-major axis? a. 0.78 days
a. b. 0.92 days
b. c. 1.08 days
c. d. 1.21 days
d.
250 Chapter 7 • Test Prep

7.2 Newton's Law of Universal Gravitation produce the same feeling? How does this demonstrate
and Einstein's Theory of General Relativity Einstein’s postulate on which he based the theory of
general relativity?
24. What did the Cavendish experiment measure?
a. It would feel the same if the force of gravity
a. The mass of Earth
suddenly became weaker. This illustrates Einstein’s
b. The gravitational constant
postulates that gravity and acceleration are
c. Acceleration due to gravity
indistinguishable.
d. The eccentricity of Earth’s orbit
b. It would feel the same if the force of gravity
25. You have a mass of and you have just landed on suddenly became stronger. This illustrates
one of the moons of Jupiter where you have a weight of Einstein’s postulates that gravity and acceleration
. What is the acceleration due to gravity, , on are indistinguishable.
the moon you are visiting? c. It would feel the same if the force of gravity
a. suddenly became weaker. This illustrates Einstein’s
b. postulates that gravity and acceleration are
c. distinguishable.
d. d. It would feel the same if the force of gravity
suddenly became stronger. This illustrates
26. A person is in an elevator that suddenly begins to Einstein’s postulates that gravity and acceleration
descend. The person knows, intuitively, that the feeling are distinguishable.
of suddenly becoming lighter is because the elevator is
accelerating downward. What other change would

Short Answer c. An ellipse is an open curve wherein the distances

from the two foci to any point on the curve are
7.1 Kepler's Laws of Planetary Motion equal.
27. Explain how the masses of a satellite and its parent body d. An ellipse is a closed curve wherein the distances
must compare in order to apply Kepler’s laws of from the two foci to any point on the curve are
planetary motion. equal.
a. The mass of the parent body must be much less 30. Mars has two moons, Deimos and Phobos. The orbit of
than that of the satellite. Deimos has a period of and an average
b. The mass of the parent body must be much greater
radius of . The average radius of the
than that of the satellite.
orbit of Phobos is . According to
c. The mass of the parent body must be equal to the
Kepler’s third law of planetary motion, what is the
mass of the satellite.
period of Phobos?
d. There is no specific relationship between the
a.
masses for applying Kepler’s laws of planetary
b.
motion.
c.
28. Hyperion is a moon of the planet Saturn. Its orbit has an d.
eccentricity of and a semi-major axis of
. How far is the center of the orbit from 7.2 Newton's Law of Universal Gravitation
the center of Saturn? and Einstein's Theory of General Relativity
a.
31. Newton’s third law of motion says that, for every action
b.
force, there is a reaction force equal in magnitude but
c.
that acts in the opposite direction. Apply this law to
d.
gravitational forces acting between the Washington
29. The orbits of satellites are elliptical. Define an ellipse. Monument and Earth.
a. An ellipse is an open curve wherein the sum of the a. The monument is attracted to Earth with a force
distance from the foci to any point on the curve is equal to its weight, and Earth is attracted to the
constant. monument with a force equal to Earth’s weight. The
b. An ellipse is a closed curve wherein the sum of the situation can be represented with two force vectors
distance from the foci to any point on the curve is of unequal magnitude and pointing in the same
constant. direction.

Access for free at openstax.org.

 Chapter 7 • Test Prep 251

b. The monument is attracted to Earth with a force as a particle or a wave, has no rest mass. Despite this
equal to its weight, and Earth is attracted to the fact gravity bends a beam of light.
monument with a force equal to Earth’s weight. The a. True
situation can be represented with two force vectors b. False
of unequal magnitude but pointing in opposite
33. The average radius of Earth is . What is
directions.
Earth’s mass?
c. The monument is attracted to Earth with a force
a.
equal to its weight, and Earth is attracted to the
b.
monument with an equal force. The situation can
be represented with two force vectors of equal c.
magnitude and pointing in the same direction. d.
d. The monument is attracted to Earth with a force 34. What is the gravitational force between two
equal to its weight, and Earth is attracted to the people sitting apart?
monument with an equal force. The situation can a.
be represented with two force vectors of equal b.
magnitude but pointing in opposite directions. c.
32. True or false—Gravitational force is the attraction of the d.
mass of one object to the mass of another. Light, either

Extended Response sun. Halley’s Comet returns once every 75.3 years. What
is the average radius of the orbit of Halley’s Comet in
7.1 Kepler's Laws of Planetary Motion AU?
35. The orbit of Halley’'s Comet has an eccentricity of 0.967 a. 0.002 AU
and stretches to the edge of the solar system. b. 0.056 AU
Part A. Describe the shape of the comet’s orbit. c. 17.8 AU
Part B. Compare the distance traveled per day when it is d. 653 AU
near the sun to the distance traveled per day when it is at
the edge of the solar system. 7.2 Newton's Law of Universal Gravitation
Part C. Describe variations in the comet's speed as it and Einstein's Theory of General Relativity
completes an orbit. Explain the variations in terms of
37. It took scientists a long time to arrive at the
Kepler's second law of planetary motion.
understanding of gravity as explained by Galileo and
a. Part A. The orbit is circular, with the sun at the
Newton. They were hindered by two ideas that seemed
center. Part B. The comet travels much farther
like common sense but were serious misconceptions.
when it is near the sun than when it is at the edge
First was the fact that heavier things fall faster than light
of the solar system. Part C. The comet decelerates
things. Second, it was believed impossible that forces
as it approaches the sun and accelerates as it leaves
could act at a distance. Explain why these ideas
the sun.
persisted and why they prevented advances.
b. Part A. The orbit is circular, with the sun at the
a. Heavier things fall faster than light things if they
center. Part B. The comet travels much farther
have less surface area and greater mass density. In
when it is near the sun than when it is at the edge
the Renaissance and before, forces that acted at a
of the solar system. Part C. The comet accelerates
distance were considered impossible, so people
as it approaches the sun and decelerates as it leaves
were skeptical about scientific theories that
the sun.
invoked such forces.
c. Part A. The orbit is very elongated, with the sun
b. Heavier things fall faster than light things because
near one end. Part B. The comet travels much
they have greater surface area and less mass
farther when it is near the sun than when it is at the
density. In the Renaissance and before, forces that
edge of the solar system. Part C. The comet
act at a distance were considered impossible, so
decelerates as it approaches the sun and accelerates
people were skeptical about scientific theories that
as it moves away from the sun.
invoked such forces.
36. For convenience, astronomers often use astronomical c. Heavier things fall faster than light things because
units (AU) to measure distances within the solar system. they have less surface area and greater mass
One AU equals the average distance from Earth to the density. In the Renaissance and before, forces that
252 Chapter 7 • Test Prep

act at a distance were considered impossible, so 7.35×1022 kg, respectively. The distance from Earth to the
people were quick to accept scientific theories that moon is 3.80×105 km. At what point between the Earth
invoked such forces. and the moon are the opposing gravitational forces
d. Heavier things fall faster than light things because equal? (Use subscripts e and m to represent Earth and
they have larger surface area and less mass density. moon.)
In the Renaissance and before, forces that act at a a. 3.42×105 km from the center of Earth
distance were considered impossible because of b. 3.80×105 km from the center of Earth
people’s faith in scientific theories. c. 3.42×106 km from the center of Earth
d. 3.10×107 km from the center of Earth
38. The masses of Earth and the moon are 5.97×1024 kg and

Access for free at openstax.org.