Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
'I GRAVITY THE BIG IDEA The same force that pulls apples off trees and pulls you toward Earth's center-gravity-underlies the motion of satellites. .6. Earth's gravity holds the space shuttle in orbit. If it didn't, the space shuttle would fly off in a straight line. I s the space shuttle in the grip of Earth gravity, or is it beyond it? Was gravity discovered by Isaac Newton? Or was gravity discovered by earlier people who fell from trees or from their caves? If Newton didn't discover gravity, what did he discover about gravity? Does gravity reach the Moon? Does it reach to the planets? To the stars? How far does gravity reach? Why are there ocean tides? Is it true that if a ba ll is thrown fast enough, and above the atmosphere, it becomes a satellite? Why doesn't Earth's gravitational pull make a satellite crash into Earth? To answer t hese questions, we need to learn the physics of gravity and satellite motion. DISCOVER! What Makes Ink Flow in a Ballpoint Pen? Observe and Record 1. Put a piece of paper flat on your desk. Write your name on it with a ballpoint pen. (A ballpoint pen has a hole in the barrel through which ink flows to the pen's point.) 2. Tape the paper to a book. Hold it over your head. 3. With the pen upside down, try to write your name on the paper. Analyze and Conclude 1. Observing Were you able to write your name with the pen upside down? 2. Predicting Would you be able to write with an upside-down pencil? Try it and see. 3. Analyzing Why is there a difference between the pen's and pencil's ability to write while upside down? 4 . Applying Would an astronaut be able to use a ballpoint pen in Earth orbit? Why or why not? 113 114 PART ONE Physics '1.1 The Falling Apple and the Falling Moon FIGURE 7.1 ... Newton realizes that Earth's gravity affects both the Moon and the apple. Uh-oh, the Moon is falling! G ... )o Legend tells us that when Newton was a young man sitting under an apple tree, he made a connection that changed the way we see the world. He saw an apple fall. Perhaps he looked up through the tree branches toward the origin of the falling apple and noticed the Moon. In any event, .I Newton had the insight to realize that the force pulling on a falling apple is the same force that pulls on the Moon. Newton realized that Earth's gravity reaches to the Moon. Why doesn't the Moon fall toward Earth, as an apple falls from a tree? If the apple or anything else drops from rest, it falls in a vertical straight-line path. To get a better idea of this, consider a tree in the back of a truck (Figure 7.2). If the truck is at rest when the apple falls, we see that the apple's path is vertical. But if the truck is moving when the apple begins its fall, the apple follows a curved path. Can you see that the faster the truck moves, the wider the curved path of the falling apple? Later in this chapter we'll see that if the apple or anything else moves fast enough so that its curved path matches Earth's curvature, it will not fall to Earth-it will fall around Earth. The falling object orbits Earth as an Earth satellite. v 0 FIGURE 7.3 ... The tangential velocity of the Moon allows it to fall around Earth rather than directly into it. FIGURE 7.2 ... If an apple falls from a tree at rest, it falls straight downward. But if it falls from a moving tree, it falls in a curved path. The Moon is an Earth satellite. As the Moon revolves around Earth, it maintains a tangential velocity-a velocity parallel to Earth's surface. Newton realized that the Moon's tangential velocity keeps it continually falling around Earth instead of directly into it. Newton further realized that the Moon's path around Earth is similar to the paths of the planets around the Sun. CHECK YOUR THINKING In Figure 7.3, we see that the Moon falls around Earth rather than straight into it. If the tangential velocity were zero, how would the Moon move? What was Newton's great insight that related a falling apple to the Moon? Answer If the Moon's tangential velocity were zero, it would fall straight down and crash into Earth! (Compare this idea with Figure 7.2.) .. CHAPTER 7 GRAVITY 115 '1.2 Newton's Law of Universal Gravitation Newton realized that everything pulls on everything else-everywhere and all the time-through the force of gravity. .I Further, Newton discovered that that force of gravity is simple--it involves only mass and distance. According to Newton, every mass attracts every other mass with a force that is directly proportional to the product of the two interacting masses. And the force is inversely proportional to the square of the distance separating them. This is the law of universal gravitation . mass 1 X_mass Force,....., _ _.:...._ _--=.2 distance 2 Expressed in symbol shorthand, where m 1 and m2 are the masses, and dis the distance between their centers. Thus, the greater the masses m 1 and m 2, the greater the force of attraction between them. The greater the distance of separation d, the weaker the force of attraction-weaker as the inverse square of the distance between their centers. Just as sheet music guides a musician playing music, equations guide an integrated science student to see how concepts are connected. CHECK YOUR THINKING 1. According to the equation for gravity, what happens to the force between two bodies if the mass of one body is doubled? 2. What happens if instead the mass of the other body is doubled? 3. What happens if the masses of both bodies are doubled? 4. What happens if the mass of one body is doubled, and the other tripled? Answers I. When one mass is doubled, the force between them doubles. 2. The force is still doubled, because it doesn't make any difference which mass doubles. (2 X 1 =I X 2; same product either way!) 3. The force is four times as much. 4. Double X triple = six:. So the force is six times as much. (If you don't see why, discuss this with a friend before going further. ) DING What two factors does the force of gravity depend on? '1.3 Gravity, Distance, and the Inverse-Square Law Gravity gets weaker with distance the same way a light gets dimmer as you move farther away from it. Consider the candle flame in Figure 7.4. Light from the flame travels in all directions in straight lines. A patch is shown 1 m from the flame. Notice that at a distance of 2 m away, the PhysicsPiace.com Video Inverse-Square Law 116 PART ONE Physics FIGURE 7.4 A Light from the flame spreads in all directions. At twice the distance, the same light is spread over four times the area; at three times the distance it is spread over nine times the area. Saying that F is inversely proportional to the square of d means, for example, that if d gets bigger by 3, F gets smaller by 9. FIGURE 7.5 ... The inverse-square law. Paint spray travels in straight lines away from the nozzle of the can. Like gravity, the "strengt h" of the spray obeys the inversesquare law. Fill in the blanks. ttC:H C:K YOUR READI G What kinds of natural phenomena does the inverse-square law apply to? light rays that fall on the patch spread to fill a patch twice as tall and twice as wide. The same light falls on a patch with four times the area. The same light 3 m away spreads to fill a patch three times as tall and three times as wide. The light would fill a patch with nine times the area. As the light spreads out, its brightness decreases. When you're twice as far away, can you see that it appears one-fo urth as bright? And can you see that when you're three times as far away, it appears one-ninth as bright? There is a rule here: The intensity of t he light decreases as the inverse square of the distance. We therefore say the inverse-square law describes the intensity of light. Paint from a paint sprayer also follows an inverse-square law. Pretend you hold a paint gun at the center of a sphere with a radius of l m (Figure 7.5). Suppose that a burst of paint produces a square patch of paint 1 mm thick. How thick would the patch be if the experiment were done in a sphere with twice the radius-that is, with the spray gun twice as far away? The answer is not half as thick, because the paint would spread to a patch twice as tall and twice as wide. It would spread over an area four times as big, and its thickness would be only-~ mm. Can you see that for a sphere of radius 3 m, the thickness of the paint patch would be only~ m m? Do you see that the thickness of paint decreases as the square of t he distance? The inverse-square law holds for light, for paint spray, and for gravity. .! The inverse-square law applies to all things that uniformly spread away from a local source throughout the surrounding space. We'll see this to be true of the electric field about an electron, light from a match, radiation from a piece of uranium, and sound from a cricket. 1 area unit Paint spray 1 layer thick 4 area units 1;4 layer thick ( ) area units ( ) area units ( ) layer thick ( ) layer thick The greater the distance from Earth's center, the less the gravitational force on an object. In using Newton's equation for gravity, the distance term d is the distance between the centers of the masses of objects attracted to each other. Note in Figure 7.6 that the girl at the top of the ladder weighs only one-fourth as much as she weighs at Earth's surface. That's because she is twice the distance from Earth's center. CHAPTER 7 GRAVITY 117 CHECK YOUR THINKING 1. How much does the force of gravity change between Earth and a receding rocket when the distance between them is doubled? Tripled? Ten times as much? 2. Consider an apple at the top of a tree. The apple is pulled by Earth's gravity with a force of 1 N. If the tree were twice as tall, would the force of gravity be on1y one-fourth as strong? Defend your answer. Answers 1 I. When the distance is doubled, the force is as much. \t\'hen tripled,~ as much. When 10 times, 1 ~ as much. 2. No, because the twice-as-tall apple tree is not twice as far from Earth's center. The taller tree would have to be 6,370 km tall (Earth's radius) for the apple's weight to reduce to~ N. For a decrease in weight by I%, an object must be raised 32 km-nearly four times the height of Mt. E\·crest. So as a practical matter we disregard the effects of everyday changes in elevation for gravity. The apple has practically the same weight at the top of the tree as at the bottom. FIGURE 7.6 A At the top of the ladder, the girl is twice as far from Earth's center and weighs only one-fourth as much as at the bottom of the ladder. So gravity gets weaker with increasing distance. But no matter how far away, Earth's gravitational force approaches, but never reaches, zero. Even if you traveled to the far reaches of the universe, the gravitational influence of home would still be with you. It may be overwhelmed by the gravitational influences of nearer and/or more massive bodies, but it is there. The gravitational influence of every material object, however small or far, is exerted through all of space. FIGURE 7 .7 A As the rocket gets farther from Earth, gravitation between the rocket and Earth decreases. Distance Apple weighs ( )N here FIGURE 7.8 A !NTERAC TIVE FIGURE, i If an apple weighs 1 N at Earth's surface, it weighs only N twice as far from Earth's center. At three times the distance, it weighs only~ N. What would it weigh at four times the distance? Five times? 118 PART ONE Physics CHECK YOUR THINKING 1. Light from the Sun, like gravity, obeys the inverse-square law. If you were on a planet twice as far from the Sun, how bright would the Sun look? 2. How bright would the Sun look if you were on a planet half as far from the Sun? Answers Just as rc is the constant for the circumference of a circle, C = reD, G is the constant for the force of gravity, m1 m2 F=GdZ. I. One-quarter as bright. 2. Four times as bright. '1.4 The Universal Constant of Gravitation, G II The law of universal gravitation can be written as an exact equation when the universal constant of gravitation, G, is used. Then we have F = Gmlm2 d2 flicHECH 0 What constant do we need to know to convert the relationship between gravitational force, mass, and distance to an exact equation? The units of G make the force come out in newtons. The magnitude of G is the sam e as the gravitation al force between two 1-kg masses that are 1 m apart: 0.0000000000667 N. G = 6.67 X 10-ll N · m 2/kg 2 The universal constant G is an extremely small number. It shows that gravity is a very weak force compared with electrical forces. The large net gravitational force we feel as weight is because of the enormity of atoms in planet Earth that are pulling on us. Comparing Gravitational Attractions How do different gravitational pulls compare? Try a few calculations with Newton's law of universal gravitation. Solutions (a) Mars: m1m2 F = G---;]2 ( 6.67 Problems A 3-kg newborn baby at Earth's surface is attracted by gravity to Earth with a force of about 30 N. (a) Calculate the force of gravity with which the baby on Earth is attracted to the planet Mars, when Mars is closest to Earth. (The mass of the Mars is 6.4 X 10 23 kg, and its closest distance from Earth is 5.6 X 10 10 m .) (b) Calculate the force of gravity between the baby and the physician who delivers her. Assume that the physician has a mass of 100 kg and is 0.5 m from the baby. (c) How do these forces compare? X 10-l l N · m 2/kg 2)(3 kg)(6.4 X 1023 kg) (5.6 X J0 10 m) 2 = 4.1 X 10- 8 N (b) Physician: mLm2 F=G-- d2 = ( 6.67 X 10- 11 = 8.0 X 10- 8 N N · m 2/kg2)(3 kg)(10 2 kg) (0.52 m ) (c) The baby is pulled toward the physician with about twice the gravitational force with which she is pulled toward Mars. .. CHAPTER 7 _ll GRAVITY 119 ,..,. INTEGRATED SCIENCE -.., EARTH SCIENCE AND ASTRONOMY -~ Ocean Tides Sailors have always known there is a connection between ocean tides and the Moon. .! Newton was the first to show that tides are caused by diHerences in the gravitational pull by the Moon on Earth's opposite sides. Because gravitational force gets weaker with distance, the gravitational force between Earth and the Moon is stronger on the side of Earth nearer to the Moon than on the opposite side of Earth. _. FIGURE 7.9 Low tide; high tide. Low tide High tide To understand why these different pulls produce tides, let's look at a ball ofJell-0 (Figure 7.10). If you exerted the same force on every part of the ball, the ball would remain perfectly round as it accelerated. But if you pull harder on one side than the other, the different pulls would stretch the ball. That's what's happening to this big ball on which we live. Different pulls of the Moon stretch Earth, most notably in its oceans. The stretch produces an average ocean bulge of nearly 1 m on each side of Earth. That's why the oceans on opposite sides of Earth bulge about 1 m above the ocean's average surface level. Earth rotates once per day, so a given point on Earth passes beneath both of these bulges each day. This produces two sets of ocean tides per day-two high tides and two low tides. The Moon pulls more on the side of Earth closest to it, and less on the opposite and farther side. This difference in pulls stretches Earth to produce ocean bulgesexperienced as tides. .. FIGURE 7.10 A ball of Jell-0 stays spherical when all parts are pulled equally in the same direction. When one side is pulled more than the other, its shape is distorted. While Earth rotates, the Moon also moves in its orbit. The Moon appears at the same position in our sky every 24 hours and 50 minutes, so the two-high-tide cycle is actually at 24-hour-and-50-minute intervals. This means that tides do not occur at the same time every day. The Sun also contributes to ocean tides, but it's about half as effective as the Moon. Interestingly, the Sun pulls 180 times as hard on Earth as on the Moon. Why aren't tides due to the Sun 180 times as large as tides due to the Moon? Because of the Sun's great distance, the 120 PART ONE Physics FIGURE 7.11 lill> Two tidal bulges are produced by differences in gravitational pulls by the Moon. 1tcHECH EADING What causes tides? 0 difference in gravitational pulls on opposite sides of Earth is very small. In other words, the Sun pulls almost as hard on the far side of Earth as it does on the near side. You'll understand tides more when you tackle "Ocean Tides" in the Conceptual Integrated Science Explorations Practice Book. CHECK YOUR THINKING 1. If you pull a blob of Jell-0 equally on all parts, it will keep its shape as it moves. But if you pull harder on one end than the other, it will stretch. How does this relate to tides? 2. If the Moon didn't exist, would Earth still have ocean tides? If so, how often? 3. We know that both the Moon and the Sun produce our ocean tides. And we know that the Moon plays the greater role because it is closer. Does its closeness mean it pulls the oceans with more gravitational force than the Sun? Answers I. Just as differences in pulls on the Jell-0 will distort it, differences in pulls on the oceans distort the ocean and produce tides. 2. Yes, Earth's tides would be due only to the Sun. They'd occur twice per day (every I 2 hours instead of every I 2.4 hours) because of Earth's daily rotation. 3. No, the Sun's pull is much stronger. But tides arc not caused by gravitational pulls. Tides are caused by differences in pulls across a body. Differences in pulls, not pulling strength, is the key to tides. UNift'ING CONCEPT Newton's Laws of Motion SECTION 2.5 PhysicsPiace.com Videos Weight and Weightlessness; Apparent Weightlessness '1.5 Weight and Weightlessness The force of gravity, like any force, can produce acceleration. Objects under the influence of gravity accelerate toward each other. Because we are almost always in contact with Earth, we think of gravity as something that presses us against Earth rather than as something that accelerates us. The pressing against Earth is the sensation we interpret as weight. Stand on a bathroom scale that is supported on a stationary floor. The gravitational force between you and Earth pulls you against the supporting floor and the scale. By Newton's third law, at the same time, the floor and scale push upward on you. Located in between you and the supporting floor are springs inside the bathroom scale. Compression of the springs is read as your weight. CHAPTER 7 INTERACTIVE fiGURl h Normal weight ~~ Qlj Less than normal weight l 121 ~ FIGURE 7.12 CJ Greater than normal weight GRAVITY The sensation of weight is equal to the force that you exert against the supporting floor. If the floor accelerates up or down, your weight seems to vary. You feel weightless when you lose your support in free fall. I Zero weight If you repeat this weighing procedure in a moving elevator, your weight reading would vary-not during steady motion, but during accelerated motion. If the elevator accelerates upward, the bathroom scale and floor push harder against your feet. So the springs inside the scale are compressed even more. The scale shows an increase in your weight. If the elevator accelerates downward, the scale shows a decrease in your weight. The support force of the floor is now less. If the elevator cable breaks and the elevator falls freely, the scale registers zero pounds. You are "weightless." Similarly, astronauts in Earth orbit are weightless because they have no support force as they fall freely around Earth. In Chapter 2, we defined weight as the gravitational force acting on an object. We can now refine this definition and say that the weight of an object is the force it exerts against a supporting floor (or weighing scale). According to this definition, you are as heavy as ~ FIGURE 7.13 These astronauts in training feel weightless because they are freely falling, like a tossed projectile. They are not pressing against anything that would provide a support force. 124 PART ONE Physics CHECK YOUR THINKING At the instant a horizontal cannon fires a cannonball from atop a high cliff, another cannonball is simply dropped from the same height. Which hits the ground below first, the one fired downrange, or the one that drops straight down? Answer Both cannonballs hit the ground at the same time, because both fall the snme vertical distance. Tutorial Projectile Motion Videos Projectile Motion Demo; More Projectile Motion In Figure 7.19, we consider a stone thrown upward at an angle. If there were no gravity, the path would be along the dashed line with the arrow. Positions of the stone at 1-s intervals along the line are shown by red dots. Because of gravity, the actual positions (dark dots) are below these points. How far below? The answer is, the same distance an object would fall if it were dropped from the red-dot positions. When we connect the dark dots to plot the path, we get a different parabola. In Figure 7.20, we consider a stone thrown at a downward angle. The physics is the same. If there were no gravity, it would follow the dashed line with the arrow. Because of gravity, it falls beneath this line, just as in the previous cases. The path is a somewhat different parabola. .. e '"' --' .' FIGURE 7.19 .... I A stone thrown at an upward angle would follow the dashed line in the absence of gravity. Because of gravity, it falls beneath this line and describes the parabola shown by the solid curve. FIGURE 7.20 .A .-,' A stone thrown at a downward angle follows a somewhat different parabola. CHAPTER 7 DISCOVER! GRAVITY 3 4 125 5 Make your own model of projectile paths. On a ruler or a stick, at position 1, hang a bead from a string 1 em long as shown. At position 2, hang a bead from a string 4 em long. At position 3, do the same with a 9-cm length of string. At position 4, use 16 em of string; for position 5, use 25 em of string. Hold the stick horizontally and you have a version of Figure 7 .18. Hold it at a slight upward angle to show a version of Figure 7.19. Hold it at a downward angle, and you have Figure 7.20. ~ The curved path of any projectile is a combination of horizontal and vertical motions. This can be shown graphically. A vector represents the projectile's overall velocity. Its length is the projectile's speed, and its direction is the direction the projectile is moving at that point in time. For example, consider the girl throwing the stone in Figure 7.21. The velocity she gives the stone is shown by the diagonal vector. Notice that this vector has horizontal and vertical components. (Read more about drawing vector components in Appendix C.) These velocity components are independent of each other. Each of them acts as if the other didn't exist. When combined, they produce the curved path. FIGURE 7.22 A FIGURE 7.21 A The velocity of the ball (diagonal vector) has vertical and horizontal components. The vertical component relates to how high the ball will go. The horizontal component relates to the horizontal range of the ball. INTERACTIVE FIGURE The velocity of a projectile at various points. Note that the vertical component changes, while the horizontal component is the same everywhere. CHECK YOUR THINKING 1. At what part of its trajectory does a projectile have minimum speed? 2. (Challenge Question) A tossed ball changes speed along its parabolic path. When the Sun is directly overhead, does the shadow of the ball across the field also change speed? Answers 1. The speed of a projectile is a minimum at the top of its path. If it is launched vertically, its speed at the top is zero. If it is projected at an angle, the vertical component of speed is zero at the top, leaving only the horizontal component. So the speed at the top is equal to the horizontal component of the projectile's velocity at any point. 2. No, because the shadow moves at constant velocity across the field, showing exactly the motion due to the horizontal component of the ball's velocity. tfc:HEC:K What two motions add together to make a projectile's curved path? 126 PART ONE / L/ Physics .. ,--- .... / ..... ' ' "'" Ideal ' path \ / Actual path__.. ;:s::;;;:szpz= FIGURE 7 .23 \ .. £2!! === \ \ \ We have considered projectile motion without air drag. You can neglect air drag for a ball you toss back and fo rth with your friends because the speed is small. But higher speed makes a difference. Air drag is a factor for high-speed projectiles. The result of air drag is that both range (horizontal distance traveled) and altitude (height) are less (Figure 7.23). ~ INTERACTIVE FIGURE. In the presence of air drag, a high-speed projectile falls short of a parabolic path. The dashed line shows an ideal path with no air drag. The solid line is the actual path. UNIJYING CONCEPT '1. 'I Fast-Moving ProjectilesSatellites Suppose a cannon fires a cannonball so fast that its curved path matches the curvature of Earth. Then, without air drag, it would be an Earth SECTION 3.6 satellite! The same would be true if you could throw a stone fast enough. Any satellite is simply a projectile moving fast ------ ----- f enough to fall continually around Earth. 5m In Figure 7.24, we see the curved paths of a stone ~ thrown horizontally at different speeds. Whatever the pitching speed, in each case the stone drops the same vertical distance in the same time. For a 1-s drop, that distance is FIGURE 7.24 ~ 5 m (perhaps by now you have made use of this fact in Lab). Throw a stone at any speed and 1 slater it falls So if you simply drop a stone from rest, it will fall 5 m in 1 s 5 m below where it would have been if there of fall. Toss the stone sideways, and in 1 s it will be 5 m were no Earth gravity. below where it would have been without gravity. v' To be an Earth satellite, the stone's horizontal velocity must be great enough for its falling distance to match Earth's curvature. Friction What horizontal velocity must a tossed stone possess to become an Earth satellite? ,.... INTEGRATED SCIENCE w..J ASTRONOMY Earth Satellites It is a geometrical fact that the surface of Earth drops a vertical distance Do es 8000 m/s (or 8 km/s) seem fast to you? If not , convert this speed to kilomete rs per hou r. You get an impressive 29,000 km/ h (18,000 milh). Fast, indeed! of 5 m for every 8000 m tangent to the surface. (A tangent to a circle or to Earth's surface is a straight line that touches the circle or surface at only one place.) What do we call a projectile that moves fast enough to travel a horizontal distance of 8000 m during 1 s? We call it a satellite. Neglecting air drag, this 8000-m/s projectile would follow the curvature of Earth. Eight thousand meters per second, or 8 km/s, is very fast. At this speed, atmospheric friction would burn up the projectile. This happens to grains of sand and other meteors that graze Earth's atmosphere, burn up, and appear as "falling stars." That is why satellites like the space shuttles are launched to altitudes higher than 150 km-to be above the atmosphere. ===== CHAPTER 7 8,000 GRAVITY ?::;:;:.9_ / ' ; I I FIGURE 7.25 A Earth's curvature drops a vertical distance of 5 m for each 8000 m tangent (not to scale). It is a common misconception that satellites orbiting at high altitudes are free from gravity. Nothing could be further from the truth. The force of gravity on a satellite 150 km above Earth's surface is nearly as great as at the surface. If there were no gravity, motion would be along a straight-line path instead of curving around Earth. High altitude puts the satellite beyond Earth's atmosphere, but not beyond Earth's gravity. As mentioned previously, Earth gravity goes on forever. It gets weaker with distance, but it never reaches zero. Satellite motion was understood by Isaac Newton. He reasoned that the Moon is simply a projectile circling Earth under gravitational attraction. This concept is illustrated in Figure 7.27, which is an actual drawing by Newton. He compared the Moon's motion to a cannonball fired from the top of a high mountain. He imagined that the mountaintop was above Earth's atmosphere, so that air drag would not slow the motion of the cannonball. If a cannonball were fired with a low horizontal speed, it would follow a curved path and soon hit Earth below. If it were fired faster, its path would be wider and it would hit a place on Earth farther away. If the cannonball were fired fast enough, Newton reasoned, the curved path would become a circle and the cannonball would circle Earth indefinitely. It would be in orbit. Newton calculated the speed for circular orbit about Earth. However, because such a cannon-muzzle velocity was dearly impossible, he did not foresee humans launching satellites. And quite likely he didn't foresee multistage rockets. .! Both the cannonball and the Moon have a tangential ("sideways") velocity, parallel to Earth's surface. This velocity is enough to ensure motion around Earth rather than into it. Without air drag to reduce speed, the Moon or any Earth satellite "falls" around and around Earth indefinitely. Similarly, the planets continually fall arow1d the Sun in closed paths. Why don't the planets crash into the Sun? They don't because of their tangential velocities. What would happen if their tangential velocities were reduced to zero? The answer is simple enough: their motion would be straight toward the Sun and they would indeed crash into it. Any objects in the solar system without enough tangential velocity have long ago crashed into the Sun. What remains is the harmony we observe. II I 127 I ' \ ' ._ __ FIGURE 7.26 ._ If the stone is thrown fast enough that its curve matches Earth's curvature, it will be a satellite. UNI~NG CONCEPT Newton's Laws of Motions SECTION 2.5 FIGURE 7.27 A A drawing by Newton showing how a faster and faster projectile could circle Earth and become a satellite. -.ticHECK Why do the Moon and other Earth satellites fall around Earth rather than into it? PhysicsPiace.com Video Circular Orbits 128 PART ONE Physics DISCOVER! Swing a bucket of water in a vertical circle, as shown by physics teacher Marshall Ellenstein. If you swing it sufficiently fast, the water won't spill. The explanation is similar to why satellites don't "fall" to Earth. Actually, both the water in the bucket and satellites are falling. The water doesn't spill at the top of the swing because the bucket swings downward at least as fast as the water falls. Similarly, a satellite doesn't get closer to Earth because it falls a distance that matches Earth's curvature. Analogies are the way to understand concepts! CHECK YOUR THINKING 1. Can we also say that a satellite stays in orbit because it is above Earth's main pull of gravity? 2. What would happen to a rocket launched vertically that remains vertical as it rises? Answer l. No, no, no! No satellite is completcly"above" Earth's gravity.lfthe satellite were not in the grip of Earth's gravity, it would not orbit and would follow instead a straight line path. 2. After the rocket reaches its highest point it would fall back to its launching site-not a good idea! Satellite TV The time it takes an Earth satellite to make a complete trip around our planet depends on its distance from Earth. At a distance of about 5.5 Earth radii, a satellite orbits the planet in 24 h. This is also the amount of time it takes for Earth to make one full spin on its axis. That's why a satellite orbiting around the equator once every 24 h stays above the same point on the ground. We say this satellite is in geosynchronous orbit-it travels along with Earth's surface below. Satellite television technology uses communications satellites that are in geosynchronous orbit to bounce television signals from space to Earth. Because a communications satellite is in geosynchronous orbit, its position relative to the receiving dish is fixed. It's always above the location it serves. Therefore, you don't need to constantly adjust your dish to stay with your satellite-just grab the remote. 'I REVIEW WORDS TO KNOW AND USE Inverse-square law A law relating the intensity of an effect to the inverse square of the distance from the cause: . Intensity ~ l . 1 distance- Law of universal gravitation Every body in the universe attracts every other body with a mutually attracting force. For two bodies, this force is directly proportional to the product of their masses and inversely proportional to the square of the distance separating them: 5. How does the brightness of light change when a point source of light is brought twice as far away? 6. At what distance from Earth is the gravitational force on an object zero? 7.4 The Universal Constant of Gravitation, G 7. What is the magnitude of gravitational force between two 1-kg bodies that are l m apart? 8. What do we call the gravitational force between Earth and your body? 7.5 Weight and Weightlessness 9. Why do you feel weightless when falling? 10. Are astronauts weightless because they are beyond Parabola The curved path followed by a projectile near Earth under the influence of gravity only. Projectile Any object that moves through the air or through space under the influence of gravity. Satellite A projectile or small body that orbits a larger body. Universal constant of gravitation, G The proportionality constant in Newton's law of universal gravitation. REVIEW QUESTIONS 7.1 The Falling Apple and the Falling Moon Earth's gravitational influence? Defend your answer. 7.6 Projectile Motion 11 . With no gravity, a horizontally moving projectile follows a straight-line path. With gravity, how far below the straight-line path does it fall compared with the distance of free fall? 12. As an object moves horizontally through air (without air drag), how much speed does it gain moving horizontally? How much vertically? 13. A ball is batted upward at an angle. What happens to the vertical component of its velocity as it rises? As it falls? 1 . What connection did Newton make between a 7.7 Fast-Moving Projectiles-Satellites falling apple and the Moon? 2. In what sense does the Moon "fall"? 14. How can a projectile "fall around Earth"? 7.2 Newton's Law of Universal Gravitation 3. State Newton's law of universal gravitation in ?. words. Then do the same with one equation. 7.3 Gravity, Distance, and the Inverse-Square Law 4. How does the force of gravity between two bodies change when the distance between them is doubled? 129 130 PART ONE Physics ,...,_ INTEGRATED SCIENCE .,_, THINK AND LINK Earth Science and AstronomyOcean Tides 1. Do tides depend more on the strength of gravitational pull or on the difference in strengths? Explain. 2. Why do both the Sun and the Moon exert a greater gravitational force on one side of Earth than the other? 3. Which pulls with greater force on Earth's oceans, the Sun or the Moon? Which is more effective in raising tides? Why are your answers different? Astronomy-Earth Satellites 1. Why will a projectile that moves horizontally at 8 km/s follow a curve that matches the curvature of Earth? 2. Why is it important that the projectile in Question 1 be above Earth's atmosphere? 3. Arc the planets of the solar system simply projectiles falling around and around the Sun? THINK AND DO 1. Hold your hands outstretched with one hand twice as far from your eyes as the other. Make a casual judgment about which hand looks bigger. Most people see them to be about the same size, while many see the nearer hand as slightly bigger. Very few people see the nearer hand as four times as big. But by the inverse-square law, the nearer hand should appear twice as tall and twice as wide. Twice times twice means four times as big! That's four times as much of your visual field as the farther hand. Your belief that your hands are the same size is so strong that you likely overrule this information. Try it again, only this time overlap your hands slightly and view them with one eye closed. Aha! Do you now more clearly see that the nearer hand appears bigger? This raises an interesting question: what other illusions do you have that are not so easily checked? 2. Repeat the eyeballing experiment, only this time use two dollar bills--one flat, and the other folded in half lengthwise and again widthwise, so it has one-fourth the area. Now hold the two in front of your eyes. Where do you bold the folded bill so that it looks the same size as the unfolded one? Share this with your friends! THINK AND EXPLAIN 1. Comment on whether this label on a consumer product should be cause for concern. CAUTION: The mass of this product pulls on every other mass in the universe, witlz an attracting force that is proportional to the product of the masses and inversely proportional to the square of the distance between them. 2. Gravitational force acts on all objects in propor- tion to their masses. Why, then, doesn' t a heavy object fall faster than a light object? 3. What would be the path of the Moon if somehow all gravitational forces on it vanished to zero? 4. Is the force of gravity stronger on a piece of iron than a piece of wood if both have the same mass? Defend your answer. 5. What is the magnitude and direction of the gravitational force that acts on a teacher who weighs 1000 N at the surface of Earth? 6. Earth and the Moon are attracted to each other by gravitational force. Does the more massive Earth attract the less massive Moon with a force that is greater, smaller, or the same as the force with which the Moon attracts Earth? 7. Would ocean tides exist if the gravitational pull of the Moon (and Sun) were somehow equal on all parts of the world? Explain. 8 . A heavy crate accidentally falls from a high-flying airplane just as it flies directly above a shiny red sports car parked in a car ~ lot. Relative to the car, ~ where will the crate crash? 9. How does the vertical component of motion for a ball kicked off a high cliff compare with the motion of vertical free fall? CHAPTER 7 1 o. In the absence of air drag, why does the horizontal component of the ball's motion not change, while the vertical component does? 11. A park ranger shoots a monkey hanging from a branch of a tree with a tranquilizing dart. The ranger aims directly at the monkey, not realizing that the dart will follow a parabolic path and thus fall below the monkey. The monkey, however, sees the dart leave the gun and lets go of the branch to avoid being hit. Will the monkey be hit anyway? Defend your answer. GRAVITY 131 are doubled but the distance between them stays the same. 3. Show that there is no change in the force of gravity between two objects when their masses arc doubled and the distance between them is also doubled. 4. The mass of Earth is 6 X 1024 kg and the mass of the Sun is 2 X I 030 kg. The average distance between the two is 1.5 X 10 11 m. Show that the force of gravity between them is 3.6 X 1022 N. 5. Students in a lab roll a steel ball off the edge of a table. They measure the speed of the horizontally launched ball to be 4.0 m/s. They also know that when the ball is simply dropped from rest off the edge of the table, it takes 0.5 s to hit the floor. Show that they should place a small piece of paper 2.0 m from the bottom of the table so that the rolled ball will hit it when it lands. 12. Because the Moon is gravitationally attracted to Earth, why doesn't it simply crash into Earth? 13. What is the effect of air drag on the height and range of a batted baseball? THINK AND COMPARE The planet and its moon gravitationally attract each other. Rank the gravitational attraction between them from most to least. A@. ~ 2d THINK AND SOLVE 1. Suppose you stood atop a ladder that was so tall that you were three times as far from Earth's center as you presently are. Show that your weight would be one-ninth of its present value. 2. Show that the gravitational force between two planets is doubled if the masses of both planets MULTIPLE CHOICE QUESTIONS Choose the best answer to the following questions. Check your answers with your teacher. 1. The force of gravity between two planets depends on their (a) masses and distance apart. (b) planetary atmospheres. (c) rotational motions. (d) All of the above 2. When the distance between two stars is reduced by!, the force between them (a) decreases by (b) decreases by~(c) increases twice times as much. (d) increases two times as much. 3. If the Sun were twice as massive, its pull on Mars would be (a) unchanged. (b) twice. (c) half. (d) four times as much. 1- 132 PART ONE Physics 4. The body most responsible for ocean tides on Earth is the Earth itself. (b) the Sun. (c) the Moon. (d) Jupiter and other massive planets. 5. The highest ocean tides occur when Earth and Moon are (a) lined up with the Sun. (b) at right angles to the Sun. (c) at any angle to the Sun. (d) lined up during spring. 6. When an astronaut in orbit is weightless, he or she is (a) beyond the pull of Earth's gravity. (b) still in the grips of Earth's gravity. (c) in the grips of interstellar gravity. (d) None of the above 7. When no air resistance acts on a fast-moving baseball, its acceleration is (a) zero. (b) due to a combination of constant horizontal motion and accelerated downward motion. (c) opposite to the force of gravity. (d) downward, g. (a) 8. When you toss a projectile sideways, it curves as it falls. lt will be an Earth satellite if the curve it makes (a) matches the curve of Earth's surface. (b) results in a straight line. (c) spirals out indefinitely. (d) None of the above 9. A satellite in geosynchronous orbit (a) takes 24 hours to complete each orbit. (b) travels at a speed of 8 km/s. (c) travels at a speed slightly greater than 8 km/s. (d) has zero orbital speed and doesn't move. 10. A satellite in Earth orbit is above Earth's (a) atmosphere. (b) gravitational field. (c) Both (a) and (b) (d) Neither (a) nor (b)