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Ch5HW - 92 Chapterj Gravity my IL;2" S U M M A R Y m3...

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Unformatted text preview: 92 Chapterj Gravity my) IL;2"_ S U M M A R Y m3:- Although no one knows what gravity is or why it exists, we can accurately describe how gravity affects the motions ol'objects. The same laws of motion work on Earth and in the heavens. Newton’s universal law of gravitation states that a gravitational attraction exists between every pair of objects and is given by my m2 15:0 2 T ‘ ‘ Studying the motions of L l f distant celestial objects, as i . 5 well as the distributions of ' . mass within multiparticle w : systems such as globular y , clusters and galaxies, pro- ‘_ : vides direct evidence that ' I 5 I gravity extends throughout l the entire visible universe. where m1 and m2 are the masses of the two objects, 7“ is the distance between their centers, and G is the gravitational constant. The value of G was first determined by Cavendish and is believed to be constant with time and space. The higher a satellite's orbit, the longer it takes to complete one orbit. A satel~ lite with a period of 1 day and positioned above the equator would appear to re- main fixed in the sky. The Moon, a natural satellite, takes 27.3 days to complete one orbit around Earth. The force of gravity can be considered constant when the motion occurs over short distances near Earth’s surface. However, small variations occur in the accel- eration due to gravity with latitude, elevation, and the types of surface material. At larger distances the force decreases as the square of the distance. Stars in binary systems revolving around each other and the motion of stars within galaxies sup— port this idea. i The value of the gravitational field at any point in space is equal to the force experienced by a lakilogram rnass placed at that point. “KEY TERMS : field: A region of space that has a number 01- a vector assigned tional to the square 01‘ the distance. l[' the distance is doubled, 1 5T tocvery point. ‘ ~ - the force decreases by a factor ol4. ' . gravitational field: The gravitational l‘Ol‘CC cxpcficnccd by a law of Universal gravitation: All IT'IEISl-iCS CXEI'L fUI‘L‘CS Ol'l Elli i 1 . likilogram mass placed at a point in space. other masses. The force F between a'ny two objects is given by ! ‘ . , _ . _ . _ F = GmlmZ/rg, where G is a universal constant, 7.”; and m! are the Inverse-square: A relationsinp in which a quantity 1s related , . .‘ . . _ _ masses of the two ObJCCILS, anti r is the (llSlElnCC between their to the reelprocal of the square ofa second quantlty, An example centers is the law of universal gravitation; the force is inversely propor- CONCEPTUAL QUESTIONS I. What force (if any) drives the planets along their orbits? 2. What force (if any) causes the planets to execute (nearly) circular orbits? @ Is the size of the gravitational force that Earth exerts on the Moon smaller than, larger than, or the same size as the force the Moon exerts on Earth? Why? I l 4. Earth exerts a gravitational force of 10,000 newtons on a satellite. What force does the satellite exert on Earth? 5. How does the average acceleration of the Moon about the Sun compare with that of Earth about the Sun? NASA Question 3 6. If an apple were placed in orbit at the same distance from Earth as the Moon, what acceleration would the apple have? . What happens to the surface area of a cube when the length ofeach side is doubled? How does this compare to what happens to the surface area of a sphere when you double its radius? . What happens to the volume of a cube if the length of each side is doubled? How does this compare to what happens to the volume of a sphere when you double its radius? .A future space traveler, Skip Parsec, lands on the planet * l 0. Astronaut Skip visits planet MSUS, which is composed of the same materials as Earth, but has twice the radius. If l2 MSU3, which has the same mass as Earth but twice the ra- dius. If Skip weighs 800 newtons on Earth’s surface, how Inuch does he weigh on MSUB's surface? Skip weighs 800 newtons on Earth’s surface, how much does he weigh on MSUB's surface? . Why didn't Newton have to know the mass of the Moon to obtain the law of universal gravitation? . Comment on the following statement made by a TV news- caster during an Apollo flight to the Moon: “The space- craft has now left the gravitational force of Earth." @ In a parallel universe, there is a planet with the same mass I4. 15. 16 [7 I8 I? and radius as Earth. However, when an apple is dropped on this planet, it falls with acceleration of20 (meters per second) per second. What is the value of Gin this parallel universe? For simplicity we use 30 (meters per second) per second for ginstead of the more accurate 9.8 (meters per second) .. per second. If Cavendish had made the same approxima- tion, would his estimate for Earth’s mass have been too ' high or too low? If a satellite in a circular orbit above Earth is continually “falling,” why doesn't it quickly return to Earth? . As a satellite orbits Earth, the gravitational force is con— stantly pulling the satellite inward. What counters this force? . Astronaut Story Musgrave has spent a total of 1281 hours, 59 minutes, and 22 seconds in space on his six space shuttle missions. If Story’s mass is 80 kilograms, then the gravitational force acting on him in orbit is approximately 730 newtons. Why did he feel weightless? . You are standing on a bathroom scale in an elevator when suddenly the cable breaks and the elevator begins to fall freely down the shaft. How does the reading on the scale change from just before to just after the cable breaks? How does the force of gravity that Earth exerts on you change over the same time interval? . NASA uses the famous “Vomit Comet," a KC-135 cargo plane, to provide astronauts and scientists a simulated zero-gravity environment. The plane flies a series of Conceptual Questions 93 parabolic arcs, as shown in the figure. Explain why the passengers feel “weightless" when the plane is near the top of its arc. 20. *2l. 22. 23- You have no doubt seen pictures ofthe astronauts floating around inside the space shuttle as it orbits some 300 kilo— meters above Earth's surface. Would the force of gravity Earth exerts on an astronaut be the same, a little less, or much less than the force that the astronaut would experi- ence on Earth‘s surface? Why? If Earth were hollow but still had the same mass and ra- dius, would your weight be different? Why? Astronomers believe that when Earth first formed, its com- position was uniform. Over time, the heavier materials sunk to the middle to create a dense iron core, with the less dense materials toward the outside. How did the value of‘gr at Earth’s surface change while this process was occurring? The gravitational force between two books sitting on a table does not cause them to accelerate toward each other because of frictional forces. If these same two books were floating near each other in deep space, they would still not appear to accelerate toward each other. Why not? (@ \Nhy do we use the form 14/: mg for the gravitational force 25. NASA on an object near Earth, but the form F = Gm1m2/1’2 when the object is far from Earth? ' ' Skylab caused quite a commotion when it returned to Earth injuly 1979. Why would it suddenly return to Earth after it had been in orbit for many years? 94 26. 27. 28. (a 30. 3!. 32. 33. 34. =85. 36. 37. 38. Q 40. Chapter 5 Gravity When the Hubble Space Telescope (HST) was originally launched by the space shuttle Discovery, its approximately circular orbit was at an altitude of about 600 kilometers. However, over the next several years, the altitude de- creased so that subsequent servicing missions were re- quired to lift the HST back into the higher orbit. What is responsible for the orbital decay? How could we determine the mass of a planet such as Venus, which has no moon? Why can we not determine the mass of the Moon by not- ing that it orbits Earth in a nearly circular orbit? What can we do to determine the Moon’s mass? Would you expect the value of g to be larger or smaller than normal over a large deposit of uranium ore? Why? You are steaming across the Atlantic Ocean in a large cruise ship. What happens to your weight as the ship leaves the deep waters of the North Atlantic and enters the shallow coastal waters of the United States? What changes would occur in the solar system if the grav- itational constant G were slowly getting larger? What do you think would happen to the Moon’s orbit if the gravitational attraction between the Moon and Earth were slowly growing stronger? Is it possible for an Earth satellite to remain “stationary" over Paris? ‘Why or why not? During the Gulf War with Iraq in 1991, a newspaper story reported that American spy satellites were in stationary orbits over Iraq, providing Continuous intelligence infor- mation. Explain why this is impossible. Assume that NASA faiis in its attempt to put a communi- cations satcllitc. into geosynchronous orbit. if the orbit is too big, what apparent motion will the satellite have as seen from the rotating Earth? Some of NASA’S Earth satellites remain above a single lo cation on Earth. Why don't these geosynchronous satel- iites fall to Earth under the influence of gravity? Newton's third law says that the gravitational force exerted on Earth by the Moon is equal to that exerted on the Moon by Earth. Why is it, then, that Earth doesn’t appear to orbit the Moon? The Sun has a profound influence on Earth’s motion. Does Earth influence the Sun's motion? Explain. en the tide is high along the American western sea— board, is the tide injapan nearer high tide or low tidePJa- pan is approximately 90 degrees west of San Francisco. When it is high tide Off the coast of Ecuador, is the tide off the coast of Indonesia, which is 180 degrees around the globe from Ecuador, nearer high tide or low tide? 4!. *42. 43. If Earth rotated once every 12 hours. how long would it be between high tides?I The Moon is observed to keep the same side facing Earth at all times. If the Moon had oceans, how much time would elapse between its high tides? Which position in the figure corresponds to the new moon? This is when the Moon is above the horizon but cannot be seen because the lit side faces away from Earth. Why are high tides higher than normal during this phase? 44. 45. 46. *47. *48. 49. Questions 43 — 46 Which position in the figure corresponds to the hill moon, which is when the Moon appears as a fully lit disk? Why are high tides higher than normal during this phase? A classmate asserts that when the Moon is in position b in the figure, the gravitational elfects of the Sun and Moon tend to cancel, producing lower than normal high tides. What is wrong with this reasoning? Which positions of the Moon in the figure correspond to the smallest difference between high tide and low tide? In The fupt‘ter Efiéct, authors John Gribbin and Stephen Piagemann claim that the additional tidal force produced when all the planets lie along one line might be enough to trigger an earthquake along the San Andreas Fault in California. What do you think about the possibility? Why would the inertia and friction ofwater cause the tides to occur after the Moon passes overhead? How does the magnitude of Earth’s gravitational field change with increasing dismnce? at 5" M G J) Exercises 95 E X E R C I S E S I. Earth's speed in its orbit about the Sun is about 30 km/s. M, The masses ofthe Moon and Earth are 7.4 X 1022 kg and 6 X What is Earth '5 'ACCEIETMiOHP 102'1 kg, respectively. The Earth—Moon distance is 3.8 X 103 2. The Moon's speed in its orbit is approximately 1 kni/s, and In. What is the size 0f the gravitational force between Earth the Earth—Moon distance is 380,000 Inn. Show that these and the M00“? Does the acceleration 0f the Moon pro- numbers yield an acceleration for the Moon that is very duced by [his force agree With the value given in the text? close to that given in the ICXE- I5. Ifan astronaut in full gear has a weight of 1200 N on Earth, ®Vhat is the acceleration due to gravity at a distance of how much Will the astronautweigh on the Moon? 2 Earth radii above Earth’s surface? I l6. The acceleration due to gravity on Titan, Saturn's largest 4. If you were located halfway between Earth and the Moon, moon, is about 1.4 m/s2. What would a 50kg scientific in- what acceleration would you have toward Earth? (Ignore strumentweigh 0“ Titan? the gravitational force Of the M0011 because it is THUCh 16:55 I 7. Mercury has a radius of about 0.38 Earth radii and a mass than Earth's.) of only 0.055 Earth masses. Estimate the acceleration due 5. A solid lead sphere ofradius 10 m (about 66 ft across!) has a / to gravity on Mercury. mass of about 57 million kg. If two ofthese spheres are float« Mars has a radius of about 053 Earth radii and a mass 0E ing in deep space With their centers 20 m apart, the gravita- " only 0.11 Earth masses. Estimate the acceleration due to tiona] attraction between the spheres is only 540 N (about gravity on Mars. 120 1b) . How large would this gravitational force be if the dis- 1 . . . , . A geosynchronous satellite orbits at a distance from Earth 5 tance between the centers Of the two spheres were tripled? ._ center of about 6.6 Earth radii and takes 24 h to go around 6. Two spacecraft in outer space attract each other with a force once. What distance (in meters) does the satellite travel in of20 N: What would the attractive force be ifthey were one- one day? What is its orbitai velocity (in m/S)? half as far apart? 20. An EEO—kg satellite orbits a distant planet with a radius of 7. How would the Sun's gravitational force on Earth change if 4000 km and a period of 280 min. From the radius and Earth had onevhalf its present mass? Would Earth’s accel- ' period, you calculate the satellites acceleration to be eration change? /; 0.56 I'D/52. What is the gravitational force on the satellite? 8. The gravitationai force between two very large metal The radius ofVenus’s Orbit is 0.72 times that of Earth’s or" spheres in outer space is 50 N. How large would this force bit. How much stronger is the Sun’s gravitational field at be if the mass ofeach sphere were cut in half? Ven'us than at Earth? 9. What is the ratio of" the gravitational force on you when you are 6400 km above Earth’s surface versus when you are / standing on the surface? (Earth’s radius is 6400 km.) ow docs Earth’s gravitational force on you differ when you are standing on Earth and when you are riding in a space shuttle 400 km above Earth’s surface? (Earth’s radius is 6400 km.) @A BED-kg sateliite experiences a gravitational force of800 N. What is the radius of the satellite ’5 orbit? What is its altitude? I2. A 600-kg geosynchronous satellite has an orbital radius of 6.6 Earth radii. What gravitational force does Earth exert on the satellite? NASA I3. What is the gravitational force between two 20-kg iron balls separated by a distance of 0.5 no? How does this compare 22. By what factor is Earth’s gravitational field reduced at a dis- with the weight of either ball? tance of5 Earth radii from the center of Earth? I i 94“) wnfo Trac® College Edition For additional readings, explore InfoTrac College Edition, your online library. Go to httpz//www.infotrac-college.corn/wadsworth and use the passcode that came on the card with your book. Try these search terms: GEOS and weather satellite, gravitational field, gravitational constant, Henry Cavendish, Isaac Newton and gravity, Johannes Kepler, universal gravitation. Nicholas Copernicus, Principia and Isaac Newton, space shuttle, Syncom, Tycho Brahe. ...
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