Chapter 13 Universal Gravitation Multiple Choice 1. A satellite circles planet Roton every 2.8 h in an orbit having a radius of 1.2 × 10 7 m. If the radius of Roton is 5.0 × 10 6 m, what is the magnitude of the free-fall acceleration on the surface of Roton? a. 31 m/s 2 b. 27 m/s 2 c. 34 m/s 2 d. 40 m/s 2 e. 19 m/s 2 2. The period of a satellite circling planet Nutron is observed to be 84 s when it is in a circular orbit with a radius of 8.0 × 10 6 m. What is the mass of planet Nutron? a. 6.2 × 10 28 kg b. 5.0 × 10 28 kg c. 5.5 × 10 28 kg d. 4.3 × 10 28 kg e. 3.7 × 10 28 kg 3. A 50-kg satellite circles planet Cruton every 5.6 h in an orbit with a radius of 12 × 10 6 m. What is the magnitude of the gravitational force on the satellite by planet Cruton? a. 63 N b. 58 N c. 68 N d. 73 N e. 50 N 4. Two stars of masses M and 6M are separated by a distance D. Determine the distance (measured from M) to a point at which the net gravitational force on a third mass would be zero. a. 0.41 D b. 0.33 D c. 0.37 D d. 0.29 D e. 0.14 D 247
Transcript
Chapter 13
Universal Gravitation
Multiple Choice 1. A satellite circles planet Roton every 2.8 h in an orbit having a radius of
1.2 × 107 m. If the radius of Roton is 5.0 × 106 m, what is the magnitude of the free-fall acceleration on the surface of Roton?
a. 31 m/s2
b. 27 m/s2
c. 34 m/s2
d. 40 m/s2
e. 19 m/s2
2. The period of a satellite circling planet Nutron is observed to be 84 s when it is in a circular orbit with a radius of 8.0 × 106 m. What is the mass of planet Nutron?
a. 6.2 × 1028 kg b. 5.0 × 1028 kg c. 5.5 × 1028 kg d. 4.3 × 1028 kg e. 3.7 × 1028 kg
3. A 50-kg satellite circles planet Cruton every 5.6 h in an orbit with a radius of 12 × 106 m. What is the magnitude of the gravitational force on the satellite by planet Cruton?
a. 63 N b. 58 N c. 68 N d. 73 N e. 50 N
4. Two stars of masses M and 6M are separated by a distance D. Determine the distance (measured from M) to a point at which the net gravitational force on a third mass would be zero.
a. 0.41 D b. 0.33 D c. 0.37 D d. 0.29 D e. 0.14 D
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248 CHAPTER 13
5. What is the magnitude of the free-fall acceleration at a point that is a distance 2R above the surface of the Earth, where R is the radius of the Earth?
a. 4.8 m/s2
b. 1.1 m/s2
c. 3.3 m/s2
d. 2.5 m/s2
e. 6.5 m/s2
6. A satellite is in a circular orbit about the Earth at an altitude at which air resistance is negligible. Which of the following statements is true?
a. There is only one force acting on the satellite. b. There are two forces acting on the satellite, and their resultant is zero. c. There are two forces acting on the satellite, and their resultant is not zero. d. There are three forces acting on the satellite. e. None of the preceding statements are correct.
7. Three 5.0-kg masses are located at points in the xy plane as shown in the figure. What is the magnitude of the resultant force (caused by the other two masses) on the mass at the origin?
y
x
30 cm30 cm
40 cm40 cm
a. 2.7 × 10–8 N b. 2.1 × 10–8 N c. 1.8 × 10–8 N d. 2.4 × 10–8 N e. 2.9 × 10–8 N
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8. Three 5.0-kg masses are located at points in the xy plane, as shown. What is the magnitude of the resultant force (caused by the other two masses) on the mass at x = 0.40 m, y = 0?
y
x
30 cm30 cm
40 cm40 cm
a. 2.2 × 10–8 N b. 1.9 × 10–8 N c. 1.4 × 10–8 N d. 1.6 × 10–8 N e. 2.5 × 10–8 N
9. Three 5.0-kg masses are located at points in the xy plane, as shown. What is the magnitude of the resultant force (caused by the other two masses) on the mass at x = 0, y = 0.30 m?
y
x
0.30 m0.30 m
0.40 m0.40 m
a. 2.6 × 10–8 N b. 2.0 × 10–8 N c. 2.9 × 10–8 N d. 2.3 × 10–8 N e. 2.1 × 10–8 N
10. What is the gravitational force on a 20-kg satellite circling the Earth (radius = 6.4 × 106 m, mass = 6.0 × 1024 kg) with a period of 5.0 h?
a. 88 N b. 55 N c. 36 N d. 98 N e. 18 N
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11. A spaceship of mass m circles a planet (mass = M) in an orbit of radius R. How much energy is required to transfer the spaceship to a circular orbit of radius 3R?
a. GmM/(2R) b. GmM/(3R) c. GmM/(4R) d. GmM/(6R) e. 3GmM/(4R)
12. A spacecraft (mass = m) orbits a planet (mass = M) in a circular orbit (radius = R). What is the minimum energy required to send this spacecraft to a distant point in space where the gravitational force on the spacecraft by the planet is negligible?
a. GmM/(4R) b. GmM/R c. GmM/(2R) d. GmM/(3R) e. 2GmM/(5R)
13. A projectile is launched from the surface of a planet (mass = M, radius = R). What minimum launch speed is required if the projectile is to rise to a height of 2R above the surface of the planet? Disregard any dissipative effects of the atmosphere.
a. 21
34
⎥⎦⎤
⎢⎣⎡
RGM
b. 21
58
⎥⎦⎤
⎢⎣⎡
RGM
c. 21
23
⎥⎦⎤
⎢⎣⎡
RGM
d. 21
35
⎥⎦⎤
⎢⎣⎡
RGM
e. 21
3 ⎥⎦⎤
⎢⎣⎡R
GM
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14. An object is released from rest at a distance h above the surface of a planet (mass = M, radius = R < h). With what speed will the object strike the surface of the planet? Disregard any dissipative effects of the atmosphere of the planet.
a. 21
)(2
⎥⎦
⎤⎢⎣
⎡+ hRR
GMh
b. 212
⎥⎦⎤
⎢⎣⎡RGM
c. 21)(2
⎥⎦⎤
⎢⎣⎡ −
RhRhGM
d. 212
⎥⎦⎤
⎢⎣⎡
+ hRGM
e. 212
⎥⎦⎤
⎢⎣⎡
+ hRGM
15. What is the kinetic energy of a 200-kg satellite as it follows a circular orbit of radius 8.0 × 106 m around the Earth? (Mass of Earth = 6.0 × 1024 kg.)
a. 5.0 × 109 J b. 1.0 × 1010 J c. 1.5 × 1010 J d. 2.0 × 1010 J e. 2.5 × 109 J
16. An object is released from rest when it is a height h above the surface of a planet of mass M and radius R. What is the speed of the object just before striking the surface of the planet? Neglect any air resistance. Let h = 4.0 × 106 m, R = 5.0 × 106 m, and M = 4.0 × 1024 kg.
a. 7.8 km/s b. 3.5 km/s c. 5.4 km/s d. 6.9 km/s e. 4.8 km/s
17. A 50-kg satellite circles the Earth in an orbit with a period of 120 min. What minimum energy is required to change the orbit to another circular orbit with a period of 180 min? (Earth: radius = 6.4 × 106 m, mass = 6.0 × 1024 kg)
a. 2.9 × 108 J b. 3.5 × 108 J c. 4.1 × 108 J d. 4.7 × 108 J e. 5.9 × 108 J
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18. Planet Roton has a mass of 4.0 × 1023 kg and a radius of 2.0 × 106 m. With what speed should a space probe be launched from the surface of Roton so as to achieve a maximum distance of 3.0 × 106 m from the center of Roton?
a. 4.2 km/s b. 3.9 km/s c. 3.0 km/s d. 3.4 km/s e. 6.0 km/s
19. Planet Zero has a mass of 5.0 × 1023 kg and a radius of 2.0 × 106 m. A space probe is launched vertically from the surface of Zero with an initial speed of 4.0 km/s. What is the speed of the probe when it is 3.0 × 106 m from Zero’s center?
a. 3.0 km/s b. 2.2 km/s c. 1.6 km/s d. 3.7 km/s e. 5.9 km/s
20. What is the escape speed from a planet of mass M and radius R if M = 3.2 × 1023 kg and R = 2.4 × 106 m?
a. 5.5 km/s b. 4.2 km/s c. 5.2 km/s d. 4.8 km/s e. 3.7 km/s
21. A satellite of mass m circles a planet of mass M and radius R in an orbit at a height 2R above the surface of the planet. What minimum energy is required to change the orbit to one for which the height of the satellite is 3R above the surface of the planet?
a. R
GmM24
b. R
GmM15
c. R
GmM12
d. R
GmM21
2
e. R
GmM5
3
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22. Planet Zero has a mass of 4.0 × 1023 kg and a radius of 2.0 × 106 m. A 10-kg space probe is launched vertically from the surface of Zero with an initial kinetic energy of 8.0 × 107 J. What maximum distance from the center of Zero is achieved by the probe?
a. 3.2 × 106 m b. 4.0 × 106 m c. 6.0 × 106 m d. 5.0 × 106 m e. 2.5 × 106 m
23. Two satellites are placed in geosynchronous orbits, orbits with a period of 24 hours, where each satellite hovers over a spot on the Earth’s equator. Satellite B has three times the mass of satellite A. What is the relationship between the magnitudes of the gravitational forces of the Earth on the two satellites?
a. FB = 1
9FA.
b. FB = 1
3FA.
c. FB = FA. d. FB = 3FA. e. FB = 9FA.
24. A satellite is placed in a geosynchronous orbit. In this equatorial orbit with a period of 24 hours, the satellite hovers over one point on the equator. Which statement is true for a satellite in such an orbit?
a. There is no gravitational force on the satellite. b. There is no acceleration toward the center of the Earth. c. The satellite is in a state of free fall toward the Earth. d. There is a tangential force that helps the satellite keep up with the rotation
of the Earth. e. The force toward the center of the Earth is balanced by a force away from
the center of the Earth.
25. Two identical planets orbit a star in concentric circular orbits in the star’s equatorial plane. Of the two, the planet that is farther from the star must have
a. the smaller period. b. the greater period. c. the smaller gravitational mass. d. the larger gravitational mass. e. the larger universal gravitational constant.
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26. Which of the following quantities is conserved for a planet orbiting a star in a circular orbit? Only the planet itself is to be taken as the system; the star is not included.
a. Momentum and energy. b. Energy and angular momentum. c. Momentum and angular momentum. d. Momentum, angular momentum and energy. e. None of the above.
27. The figure below shows a planet traveling in a counterclockwise direction on an elliptical path around a star located at one focus of the ellipse. When the planet is at point A,
STAR
A
a. its speed is constant. b. its speed is increasing. c. its speed is decreasing. d. its speed is a maximum. e. its speed is a minimum.
28. The figure below shows a planet traveling in a clockwise direction on an elliptical path around a star located at one focus of the ellipse. When the planet is at point A,
STAR
A
a. its speed is constant. b. its speed is increasing. c. its speed is decreasing. d. its speed is a maximum. e. its speed is a minimum.
29. The figure below shows a planet traveling in a counterclockwise direction on an elliptical path around a star located at one focus of the ellipse. When the planet is at point A,
STARA
a. its speed is constant. b. its speed is increasing. c. its speed is decreasing. d. its speed is a maximum. e. its speed is a minimum.
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30. The figure below shows a planet traveling in a counterclockwise direction on an elliptical path around a star located at one focus of the ellipse. When the planet is at point A,
STAR
A
a. its speed is constant. b. its speed is increasing. c. its speed is decreasing. d. its speed is a maximum. e. its speed is a minimum.
31. The figure below shows a planet traveling in a clockwise direction on an elliptical path around a star located at one focus of the ellipse. When the planet is at point A,
STAR
A
a. its speed is constant. b. its speed is increasing. c. its speed is decreasing. d. its speed is a maximum. e. its speed is a minimum.
32. The figure below shows a planet traveling in a counterclockwise direction on an elliptical path around a star located at one focus of the ellipse. When the planet is at point A,
STAR
A
a. its speed is decreasing. b. its angular momentum is increasing. c. the gravitational force does no work on the planet.. d. all of the above are correct. e. none of the above is correct.
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33. The period of oscillation of an object in a frictionless tunnel running through the Earth is 84.3 min. What is the period of oscillation of an object in a similar tunnel on the Moon? (RE = 6.37 × 106 m; RM = 1.74 × 106 m; ME = 5.98 × 1024 kg; MM = 7.36 × 1022 kg.)
a. 6.03 × 10-3 min. b. 0.713 min. c. 84.3 min. d. 108.5 min. e. 139.6 min.
34. Three galaxies, each of mass M = 4.0 ×1041 kg , lie in a plane at the corners of an equilateral triangle with sides of 5.0 ×1022 m length. The magnitude of the force the other two galaxies exert on each galaxy is
a. . 4.3×1027 Nb. . 6.4 ×1027 Nc. . 7.4 ×1027 Nd. . 8.6 ×1027 Ne. . 4.3×1028 N
35. Knowing that g = 9.80 ms2 at sea level and that RE = 6.37 ×106 m , we find that
the value of g in m
s2 at a distance RE from the surface of the Earth is
a. 1.23. b. 2.45. c. 4.90. d. 7.35. e. 9.80.
36. When two solid spheres of the same material and same radius r are in contact, the magnitude of the gravitational force each exerts on the other is directly proportional to
a. r . b. r2 . c. r3 . d. r4 . e. r6 .
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37. Huyghens claimed that near the surface of the Earth the velocity downwards of an object released from rest, , was directly proportional to the square root of
the distance it had fallen,
vy
vy = c y .This is true if c is equal to
a. g4
.
b. g2
.
c. g . d. 2g . e. . 4g
38. Suppose the gravitational force of the Earth on a body was F = −KMEm
r3 . What
escape velocity v would a body need to escape the gravitational field of the Earth?
e
a. ve =KME
2RE2 .
b. ve =KME
RE2 .
c. ve =KME
2RE
.
d. ve =KME
RE
.
e. ve = KME .
39. In an isolated system of two bodies that exert gravitational forces on one another, the quantity (quantities) that remain(s) constant is(are)
a. the total energy of the system. b. the total angular momentum of the system. c. the angular positions of the two bodies. d. all of the above. e. only (a) and (b) above.
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40. Carla and Jenny are arguing about whether or not it is possible to escape the gravitational field of the Earth. Carla shows Jenny the system below where mass m is rE (not the Earth’s radius) distant from Earth and rP (not planet P’s radius) distant from planet P. Carla states that the mass m has escaped if
. Which one, if either, is correct, and why? FP on m = −FE on M
a. Carla, because the total gravitational force on m is zero at that point. b. Carla, because there is no gravitational force from Earth on m at that point. c. Carla, because there is no gravitational force on m from Earth when r > rE . d. Jenny, because there is a gravitational force on m from Earth no matter how
great the distance from the Earth. e. Jenny, because the gravitational force from the Earth can only be blocked by
a body that is larger than the Earth.
Open-Ended Problems 41. Isaac Newton was able to estimate a value for G, the universal gravitational
constant, from the following data: the radius of the Earth is about 6400 km, the average density of rocks is about 5.5 g/cm3, and g = 9.8 m/s2 near the surface of the Earth. What value did Newton obtain for G?
42. At the moment of a total eclipse, the moon lies along a line from the Earth to the sun. If your normal weight is 600 N, how much is your weight decreased by the combined pull of the sun and moon?
MSUN = 2.0 × 1030 kg, rS-E = 1.5 × 108 km
MMOON = 7.4 × 1022 kg, rM-E = 3.8 × 105 km
43. When a falling meteor is at a distance above the Earth’s surface of 3 times the Earth’s radius, what is its acceleration due to the Earth’s gravity?
44. The planet Venus requires 225 days to orbit the sun, which has a mass M = 1.99 × 1030 kg, in an almost circular trajectory. Calculate the radius of the orbit and the orbital speed of Venus as it circles the sun.
45. Imagine a hole is drilled down to the center of the Earth. A small mass m is dropped into the hole. Ignoring the Earth’s rotation, and all sources of friction, find the speed of the mass just as it reaches the Earth’s center. (ME = 6.0 × 1024 kg; RE = 6.4 × 106 m.)
46. Calculate the Earth’s angular momentum in the approximation that treats the Earth’s orbit around the sun as a circle. (MSun = 1.99 × 1030 kg; T = 3.156 × 107 s; ME = 5.98 × 1024 kg.)
