Objectives

Explain how Newton’s law of universal gravitation accounts for various phenomena, including satellite and planetary orbits, falling objects, and the tides.

Apply Newton’s law of universal gravitation to solve problems.

Key Termgravitational force

Gravitational ForceEarth and many of the other planets in our solar system travel in nearly circular orbits around the sun. Thus, a centripetal force must keep them in orbit. One of Isaac Newton’s great achievements was the realization that the centripetal force that holds the planets in orbit is the very same force that pulls an apple toward the ground—gravitational force.

Orbiting objects are in free fall.To see how this idea is true, we can use a thought experiment that Newton developed. Consider a cannon sitting on a high mountaintop, as shown in Figure 2.1. The path of each cannonball is a parabola, and the horizontal distance that each cannonball covers increases as the cannonball’s initial speed increases. Newton realized that if an object were projected at just the right speed, the object would fall down toward Earth in just the same way that Earth curved out from under it. In other words, it would orbit Earth. In this case, the gravitational force between the cannonball and Earth is a centripetal force that keeps the cannonball in orbit. Satellites stay in orbit for this same reason. Thus, the force that pulls an apple toward Earth is the same force that keeps the moon and other satellites in orbit around Earth. Similarly, a gravitational attraction between Earth and our sun keeps Earth in its orbit around the sun.

Newton’s Law of Universal Gravitation

gravitational force the mutual force of attraction between particles of matter

Newton’s Thought Experiment Each successive cannonball has a greater initial speed, so the horizontal distance that the ball travels increases. If the initial speed is great enough, the curvature of Earth will cause the cannonball to continue falling without ever landing.

FIGURE 2.1

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Differentiated Instruction

Preview VocabularyScientific Meanings Parabola is a geometric term that is widely used in many scientific and technical fields. A parabola is an open curve generated by intersecting a plane with a cone, such that the plane is parallel to a plane passing through the altitude. The intersecting part of these two shapes is called a parabola.

FIGURE 2.1 Point out that the nature of the cannonball’s path depends as much on the cannonball’s tangential velocity as on the strength of the gravitational force. Have students compare the path of the cannonball when the ball’s tangential speed is low with the paths of the projectiles that were studied in the chapter on two-dimensional motion.

Ask How is the motion of an orbiting object similar to the motion of a projectile with two-dimensional motion?

Answer: The orbiting object has a constant tangential speed, just as the horizontal component of a projectile’s speed is constant.

Ask What happens when the speed of the projectile is so great that the force required to keep the projectile moving in a circle is larger than the gravitational force acting on the projectile?

Answer: The projectile continues on a curved path, moving farther away from Earth’s center as the force due to gravity weakens.

� Plan and Prepare

� Teach

TEACH FROM VISUALS

InclUSIonVisual learners will benefit from a demonstration of the cannonball example shown in Figure 2.1. Make a ramp on the desktop using a pile of books as a support, and roll a steel ball down it. Let the ball roll off the desk and onto the floor. Changing the height of the ramp will change the speed of the ball as it leaves the desk and therefore the distance that the ball travels before it hits the ground.

230 Chapter 7

SEcTIon 2

F F=mE c

F F=Em c

Gravitational force depends on the masses and the distance.Newton developed the following equation to describe quantitatively the magnitude of the gravitational force if distance r separates masses m1 and m2 :

Newton’s Law of Universal Gravitation

Fg = G m1m2 _

r 2

gravitational force = constant × mass 1 × mass 2 ___ (distance between masses)2

G is called the constant of universal gravitation. The value of G was unknown in Newton’s day, but experiments have since determined the value to be as follows:

G = 6.673 × 10 −11 N•m2 _

kg2

Newton demonstrated that the gravitational force that a spherical mass exerts on a particle outside the sphere would be the same if the entire mass of the sphere were concentrated at the sphere’s center. When calculating the gravitational force between Earth and our sun, for exam-ple, you use the distance between their centers.

Gravitational force acts between all masses.Gravitational force always attracts objects to one another, as shown in Figure 2.2. The force that the moon exerts on Earth is equal and opposite to the force that Earth exerts on the moon. This relationship is an example of Newton’s third law of motion. Also, note that the gravitational forces shown in Figure 2.2 are centripetal forces. Also, note that the gravitational force shown in Figure 2.2 that acts on the moon is the centripetal force that causes the moon to move in its almost circular path around Earth. The centripetal force on Earth, however, is less obvious because Earth is much more massive than the moon. Rather than orbiting the moon, Earth moves in a small circular path around a point inside Earth.

Gravitational force exists between any two masses, regardless of size. For instance, desks in a classroom have a mutual attraction because of gravitational force. The force between the desks, however, is negligibly small relative to the force between each desk and Earth because of the differences in mass.

If gravitational force acts between all masses, why doesn’t Earth accelerate up toward a falling apple? In fact, it does! But, Earth’s acceleration is so tiny that you cannot detect it. Because Earth’s mass is so large and acceleration is inversely proportional to mass, the Earth’s acceleration is negligible. The apple has a much smaller mass and thus a much greater acceleration.

Gravitational Force The gravitational force attracts Earth and the moon to each other. According to Newton’s third law, FEm = −FmE.

FIGURE 2.2

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FIGURE 2.2 Be sure students understand that the moon and Earth each have a centripetal force acting on them, but that the effects of the centripetal force on each is very different.

Ask What might the effect on Earth’s orbit be if the moon were as massive as Earth?

Answer: The moon and Earth would orbit around each other.

TEACH FROM VISUALS

BElow lEVElTo emphasize the concept discussed in the last paragraph of this page, ask students to calculate the acceleration of Earth toward an apple and to use the weight of the apple as the force of attraction. Give the following values for the calculation:m

apple = 0.15 kg

mEarth

= 5.97 × 1024 kgr

Earth = 6.38 × 106 m

Answer: aEarth

= 2.5 × 10−25 m/s2

Circular Motion and Gravitation 231

Gravitational Force

Sample Problem C Find the distance between a 0.300 kg billiard ball and a 0.400 kg billiard ball if the magnitude of the gravitational force between them is 8.92 × 10 -11 N.

ANALYZE Given: m1 = 0.300 kg

m2 = 0.400 kg

Fg = 8.92 × 10 −11 N

Unknown: r = ?

SOLVE Use the equation for Newton’s law of universal gravitation, and solve for r.

Fg = G m 1 m 2

_ r 2

r = √ ���� G

m1m2 _ Fg

r = √ ����������������� (6.673 × 10 –11 N•m2

_ kg2

) × (0.300 kg)(0.400 kg)

__ 8.92 × 10 –11 N

r = 3.00 × 10 –1 m

1. What must be the distance between two 0.800 kg balls if the magnitude of the gravitational force between them is equal to that in Sample Problem C?

2. Mars has a mass of about 6.4 × 1023 kg, and its moon Phobos has a mass of about 9.6 × 1015 kg. If the magnitude of the gravitational force between the two bodies is 4.6 × 1015 N, how far apart are Mars and Phobos?

3. Find the magnitude of the gravitational force a 66.5 kg person would experience while standing on the surface of each of the following bodies:

Celestial Body Mass Radius

a. Earth 5.97 × 1024 kg 6.38 × 106 m

b. Mars 6.42 × 1023 kg 3.40 × 106 m

c. Pluto 1.25 × 1022 kg 1.20 × 106 m

PREMIUM CONTENT

Interactive DemoHMDScience.com

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Problem Solving

classroom PracticeGRAVITATIonAl FoRcEFind the gravitational force that Earth (mass = 5.97 × 1024 kg) exerts on the moon (mass = 7.35 × 1022 kg) when the distance between them is 3.84 × 108 m.

Answer: 1.99 × 1020 N

PRoBlEM GUIDE cUse this guide to assign problems.

SE = Student Edition Textbook

Pw = Sample Problem Set I (online)

PB = Sample Problem Set II (online)

Solving for:

r SE Sample, 1–2; Ch. Rvw. 18–19

Pw 4–5

PB 7–10

m Pw Sample, 1–3

PB 4–6

FgSE 3; Ch. Rvw. 40

Pw 6–7

PB Sample, 1–3

*challenging Problem

AnswersPractice C 1. 0.692 m

2. 9.4 × 106 m

3. a. 651 N

b. 246 N

c. 38.5 N

� Teach continued

DEconSTRUcTInG PRoBlEMSHave students rearrange the equation to solve for distance before starting the problem.

Fg = G

m1m

2 _ r2

r2 Fg = Gm

1m

2

r2 = G m

1m

2 _ F

g

r = √ G

m1m

2 _ F

g

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Black Holes

This artist’s conception shows a disk of material orbiting a black hole. Such disks provide indirect evidence of black holes within our own galaxy.

This image from NASA’s Chandra X-ray Observatory is of Sagittarius A*, which is a supermassive black hole at the center of our galaxy. Astronomers are studying

the image to learn more about Sagittarius A* and about black holes in the centers of other galaxies.

A black hole is an object that is so massive that nothing, not even light, can escape the pull of its gravity. In 1916, Karl Schwarzschild was the first

person to suggest the existence of black holes. He used his solutions to Einstein’s general-relativity equations to explain the properties of black holes. In 1967, the physicist John Wheeler coined the term black hole to describe these objects.

In order for an object to escape the gravitational pull of a planet, such as Earth, the object must be moving away from the planet faster than a certain threshold speed, which is called the escape velocity. The escape velocity at the surface of Earth is about 1.1 × 104 m/s, or about 25 000 mi/h.

The escape velocity for a black hole is greater than the speed of light. And, according to Einstein’s special theory of relativity, no object can move at a speed equal to or greater than the speed of light. Thus, no object that is within a certain distance of a black hole can move fast enough to escape the gravitational pull of the black hole. That distance, called the Schwarzschild radius, defines the edge, or horizon, of a black hole.

Newton’s laws say that only objects with mass can be subject to forces. How can a black hole trap light if light has no mass? According to Einstein’s general theory of relativity, any object with mass bends the fabric of space and time itself. When an object that has mass or even when a ray of light passes near another object, the path of the moving object or ray curves because space-time itself is curved. The curvature is so great inside a black hole that the path of any light that might be emitted from the black hole bends back toward the black hole and remains trapped inside the horizon.

Because black holes trap light, they cannot be observed directly. Instead, astronomers must look for indirect evidence of black holes. For example, astronomers have observed stars orbiting very rapidly around the centers of some galaxies. By measuring the speed of the orbits, astronomers can calculate the mass of the dark object—the black hole—that must be at the galaxy’s center. Black holes at the centers of galaxies typically have masses millions or billions of times the mass of the sun.

The figure above shows a disk of material orbiting a black hole. Material that orbits a black hole can move at such high speeds and have so much energy that the material emits X rays. From observations of the X rays coming from such disks, scientists have discovered several black holes within our own galaxy.

WHY IT MATTERS

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why It MattersBlAck HolESA black hole is an object whose escape velocity is greater than the speed of light. Escape velocity is independent of the mass of the escaping object. Understanding how a black hole can trap light, which has no mass, requires an appeal to Einstein’s general theory of relativity. You can construct a simple model to help students grasp the difficult concept of gravity as a curva-ture of space-time. Attach a sheet of flexible rubber to a hoop, and then place a heavy object in the center of the sheet. The sheet will bend downward into a cone. You can then roll a coin in an orbit around the object in the center. Point out to students that this is only a rough three-dimensional model, while the curvature of space-time is four-dimensional.

The extreme bending of space-time that prevents light from escaping the Schwarzschild radius of a black hole occurs in a less extreme form around stars that are less compact than black holes. During a solar eclipse in 1922, astronomers observed light, which came from a distant star, bend as the light passed near the sun. This observation was one of the first pieces of evidence to support Einstein’s general theory of relativity. Direct students who wish to learn more about general relativity to the feature “General Relativity.”

Circular Motion and Gravitation 233

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Applying the Law of GravitationFor about six hours, water slowly rises along the shoreline of many coastal areas and culminates in a high tide. The water level then slowly lowers for about six hours and returns to a low tide. This cycle then repeats. Tides take place in all bodies of water but are most noticeable along seacoasts. In the Bay of Fundy, shown in Figure 2.3, the water rises as much as 16 m from its low point. Because a high tide happens about every 12 hours, there are usually two high tides and two low tides each day. Before Newton developed the law of universal gravitation, no one could explain why tides occur in this pattern.

Newton’s law of gravitation accounts for ocean tides.High and low tides are partly due to the gravitational force exerted on Earth by its moon. The tides result from the difference between the gravitational force at Earth’s surface and at Earth’s center. A full explanation is beyond the scope of this text, but we will briefly examine this relationship.

The two high tides take place at locations on Earth that are nearly in line with the moon. On the side of Earth that is nearest to the moon, the moon’s gravitational force is greater than it is at Earth’s center (because gravitational force decreases with distance). The water is pulled toward the moon, creating an outward bulge. On the opposite side of Earth, the gravitational force is less than it is at the center. On this side, all mass is still pulled toward the moon, but the water is pulled least. This creates another outward bulge. Two high tides take place each day because when Earth rotates one full time, any given point on Earth will pass through both bulges.

The moon’s gravitational force is not the only factor that affects ocean tides. Other influencing factors include the depths of the ocean basins, Earth’s tilt and rotation, and friction between the ocean water and the ocean floor. The sun also contributes to Earth’s ocean tides, but the sun’s effect is not as significant as the moon’s is. Although the sun exerts a much greater gravitational force on Earth than the moon does, the difference between the force on the far and near sides of Earth is what affects the tides.

Did YOU Know?

When the sun and moon are in line, the combined effect produces a greater-than-usual high tide called a spring tide. When the sun and moon are at right angles, the result is a lower-than-normal high tide called a neap tide. During each revolution of the moon around Earth there are two spring tides and two neap tides.

High and Low Tides Some of the world’s highest tides occur at the Bay of Fundy, which is between New Brunswick and Nova Scotia, Canada. These photographs show a river outlet to the Bay of Fundy at low and high tide.

FIGURE 2.3

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Differentiated Instruction

Teaching TipTell students that the time at which high and low tides occur at a given location is not the same each day. The times vary because the tidal period is actually 12 hours and 25 minutes. There are two high and low tides every 24 hours and 50 minutes.

Teaching TipPoint out that tidal forces are exerted on all substances and that the amount of distortion depends on the elasticity of the body under gravitational influ-ence. The rocky surfaces of the moon and Earth also experience tidal forces, but the bulge is not as great as with Earth’s oceans because rock is not as easily extended.

If an orbiting body moves too close to a more massive body, the tidal force on the orbiting body may be large enough to break the body apart. The distance at which tidal forces can become destructive is called Roche’s limit, after the 19th-century French mathematician Edouard Roche.

� Teach continued

PRE-APExpand on the fact that tidal force arises from the difference between the gravitational forces at Earth’s near surface, center, and far surface, and mention that this differential force is an inverse-cubed quantity. That is to say, as the distance r between two bodies increases, the tidal force decreases as 1 __

r3 . For this reason,

the sun has less effect on the Earth’s ocean tides than the moon does, even though it has a much greater mass than the moon does.

234 Chapter 7

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(a) (b)

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Cavendish finds the value of G and Earth’s mass.In 1798, Henry Cavendish conducted an experiment that determined the value of the constant G. This experiment is illustrated in Figure 2.4. As shown in Figure 2.4(a), two small spheres are fixed to the ends of a suspended light rod. These two small spheres are attracted to two larger spheres by the gravitational force, as shown in Figure 2.4(b). The angle of rotation is measured with a light beam and is then used to determine the gravitational force between the spheres. When the masses, the distance between them, and the gravitational force are known, Newton’s law of universal gravitation can be used to find G. Once the value of G is known, the law can be used again to find Earth’s mass.

Gravity is a field force.Newton was not able to explain how objects can exert forces on one another without coming into contact. His mathematical theory described gravity, but didn’t explain how it worked. Later work also showed that Newton’s laws are not accurate for very small objects or for those moving near the speed of light. Scientists later developed a theory of fields to explain how gravity and other field forces operate. According to this theory, masses create a gravitational field in space. A gravitational force is an interaction between a mass and the gravitational field created by other masses.

When you raise a ball to a certain height above Earth, the ball gains potential energy. Where is this potential energy stored? The physical properties of the ball and of Earth have not changed. However, the gravitational field between the ball and Earth has changed since the ball has changed position relative to Earth. According to field theory, the gravitational energy is stored in the gravitational field itself.

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. GRAVITATIONAL FIELD STRENGTHYou can attach a mass to a spring scale to find the gravita-tional force that is acting on that mass. Attach various combinations of masses to the hook, and record the force in each case. Use your data to calculate the gravitational field strength for each trial (g = Fg /m). Be sure that your calculations account for the mass of the hook. Average your values to find the gravita-tional field strength at your location on Earth’s surface. Do you notice anything about the value you obtained?

MATERIALS spring scale•

hook (of a known mass)•

various masses•

Gravity Experiment Henry Cavendish used an experiment similar to this one to determine the value of G.

FIGURE 2.4

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Misconception Alert!Students may not understand why gravitational field strength is equal to the force divided by the mass acted upon, rather than just the gravitational force. Explain that for a given location, the strength associated with the gravi ta tional field must be constant but that the gravitational force exerted on two different masses at the same location will differ as the masses themselves differ.

TEAcHER’S noTESAsk students about the significance of their resulting values. If results are not close to free-fall acceleration, you can find a class average. At this point, students should realize that the value of g equals the value of free-fall acceleration. Tell students that they will learn more about this later in the section.

The language of PhysicsThe symbol g represents two different quantities: free-fall acceleration on Earth’s surface (9.81 m/s2) and gravita-tional field strength. Free-fall accelera-tion at any location (a

g) is always equal

to the gravitational field strength (g) at that location.

QuickLab

Circular Motion and Gravitation 235

Conceptual Challenge

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At any point, Earth’s gravitational field can be described by the gravi ta tional field strength, abbreviated g. The value of g is equal to the magnitude of the gravitational force exerted on a unit mass at that point, or g = Fg/m. The gravitational field (g) is a vector with a magnitude of g that points in the direction of the gravitational force.

Gravitational field strength equals free-fall acceleration.Consider an object that is free to accelerate and is acted on only by gravita-tional force. According to Newton’s second law, a = F/m. As seen earlier, g is defined as Fg/m, where Fg is gravitational force. Thus, the value of g at any given point is equal to the acceleration due to gravity. For this reason, g = 9.81 m/s2 on Earth’s surface. Although gravitational field strength and free-fall acceleration are equivalent, they are not the same thing. For instance, when you hang an object from a spring scale, you are meas uring gravitational field strength. Because the mass is at rest (in a frame of reference fixed to Earth’s surface), there is no measurable acceleration.

Figure 2.5 shows gravitational field vectors at different points around Earth. As shown in the figure, gravitational field strength rapidly decreases as the distance from Earth increases, as you would expect from the inverse-square nature of Newton’s law of universal gravitation.

Weight changes with location.In the chapter about forces, you learned that weight is the magnitude of the force due to gravity, which equals mass times free-fall acceleration. We can now refine our definition of weight as mass times gravitational field strength. The two definitions are mathematically equivalent, but our new definition helps to explain why your weight changes with your location in the universe.

Newton’s law of universal gravitation shows that the value of g depends on mass and distance. For example, consider a tennis ball of mass m. The gravitational force between the tennis ball and Earth is as follows:

Fg = GmmE _

r 2

Combining this equation with the definition for gravitational field strength yields the following expression for g:

g = Fg

_ m = GmmE _

mr 2 =

GmE _ r 2

This equation shows that gravitational field strength depends only on mass and distance. Thus, as your distance from Earth’s center increases, the value of g decreases, so your weight also decreases. On the surface of any planet, the value of g, as well as your weight, will depend on the planet’s mass and radius.

Earth’s Gravitational Field The gravitational field vectors represent Earth’s gravitational field at each point. Note that the field has the same strength at equal distances from Earth’s center.

FIGURE 2.5

Gravity on the Moon The magnitude of g on the moon’s surface is about 1 __

6 of the value

of g on Earth’s surface. Can you infer from this relationship that the moon’s mass is 1 __

6 of

Earth’s mass? Why or why not?

Selling Gold A scam artist hopes to make a profit by buy-ing and selling gold at different altitudes for the same price per weight. Should the scam artist buy or sell at the higher altitude? Explain.

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Differentiated Instruction

AnswersConceptual Challenge 1. no; The distance from the center of

the moon to the moon’s surface is less than the distance between Earth’s center and Earth’s surface. These distances, along with the masses of the two bodies, affect gravitational field strength.

2. The gold should be purchased at a high altitude, where the gold weighs less, and sold at a low altitude, where the gold weighs more. (Even so, this scheme is impractical, in part because of transportation costs.)

� Teach continued

PRE-APAsk students if they have heard about the “weightlessness” of astronauts in space. Have them think about the exactness of this statement. Can astronauts reach absolute weightlessness? Explain that astronauts never reach a point at which they have no weight. As astronauts move farther away from Earth’s center, their weight will decrease until it is very small. No matter where any object is located in the universe, however, it is always being

acted on by the gravity of some other object. Even though the gravitational effect of a distant object may be minuscule, total gravitation never equals zero. For this reason, many scientists prefer the term microgravity over weightlessness.

Ask students if they have ever felt a brief sensation of weightlessness in an elevator or when a car goes over a bump. This sensation is due not to the absence of gravity, but to the absence of a force balancing gravity.

236 Chapter 7

Reviewing Main Ideas

1. Explain how the force due to gravity keeps a satellite in orbit.

2. Is there gravitational force between two students sitting in a classroom? If so, explain why you don’t observe any effects of this force.

3. Earth has a mass of 5.97 × 1024 kg and a radius of 6.38 × 106 m, while Saturn has a mass of 5.68 × 1026 kg and a radius of 6.03 × 107 m. Find the weight of a 65.0 kg person at the following locations:a. on the surface of Earthb. 1000 km above the surface of Earthc. on the surface of Saturnd. 1000 km above the surface of Saturn

4. What is the magnitude of g at a height above Earth’s surface where free-fall acceleration equals 6.5 m/s2?

Critical Thinking

5. Suppose the value of G has just been discovered. Use the value of G and an approximate value for Earth’s radius (6.38 × 106 m) to find an approxi-mation for Earth’s mass.(b

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Gravitational mass equals inertial mass.Because gravitational field strength equals free-fall acceleration, free-fall ac cel era tion on the surface of Earth likewise depends only on Earth’s mass and radius. Free-fall acceleration does not depend on the falling object’s mass, because m cancels from each side of the equation, as shown on the previous page.

Although we are assuming that the m in each equation is the same, this assumption was not always an accepted scientific fact. In Newton’s second law, m is sometimes called inertial mass because this m refers to the property of an object to resist acceleration. In Newton’s gravitation equation, m is sometimes called gravitational mass because this m relates to how objects attract one another.

How do we know that inertial and gravitational mass are equal? The fact that the acceleration of objects in free fall on Earth’s surface is always the same confirms that the two types of masses are equal. A more massive object experiences a greater gravitational force, but the object resists acceleration by just that amount. For this reason, all masses fall with the same acceleration (disregarding air resistance).

There is no obvious reason why the two types of masses should be equal. For instance, the property of electric charges that causes them to be attracted or repelled was originally called electrical mass. Even though this term has the word mass in it, electrical mass has no connection to gravitational or inertial mass. The equality between inertial and gravitational mass has been continually tested and has thus far always held up.

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SECTION 2 FORMATIVE ASSESSMENT

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Answers to Section Assessment

The language of PhysicsThe difference between inertial mass and gravitational mass can be under-stood by the ways in which they are measured. Gravitational mass is a measurement of the weight of an object (as weighed on a spring scale) divided by the gravitational field strength at the location of the measurement. The gravitational mass is pulled by a field force but does not accelerate. By contrast, inertial mass is accelerated, as in an Atwood’s machine, where the inertial mass accelerates under the force of another falling mass.

Assess Use the Formative Assessment on this page to evaluate student mastery of the section.

Reteach For students who need additional instruction, download the Section Study Guide.

Response to Intervention To reassess students’ mastery, use the Section Quiz, available to print or to take directly online at HMDScience.com.

� Assess and Reteach

1. A satellite moves tangentially around a planet and would continue to move in a straight line if there were no gravita-tional force. The combination of the acceleration due to gravity and the tangential speed causes the satellite to follow a path that is parallel to the curvature of the planet.

2. yes; The magnitude of the force is extremely small because the masses of the students are small relative to Earth’s mass.

3. a. 636 N

b. 475 N

c. 678 N

d. 656 N

4. 6.5 m/s2

5. 5.98 × 1024 kg

Circular Motion and Gravitation 237