21st Century Astronomy The Solar System Fifth Edition By Kay -Palen -Test Bank

21st Century Astronomy The Solar System Fifth Edition By Kay -Palen -Test Bank

 

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Sample Questions Are Posted Below

 

Chapter 4: Gravity and Orbits

Learning Objectives

Define the bold-faced vocabulary terms within the chapter.

Multiple Choice: 3, 4, 7, 9, 14, 15, 16, 17, 18, 47

Short Answer: 4, 5

4.1 Gravity Is a Force Between Any Two Objects due to Their Masses

Differentiate gravity, mass, and weight.

Multiple Choice: 1, 2, 10, 12

Describe the behavior of the gravitational force using Newton’s universal law of gravitation.

Multiple Choice: 5, 6

Short Answer: 2

Use Newton’s universal law of gravitation to quantify the force of gravity between two objects in different physical situations.

Multiple Choice: 11, 13

Explain how gravitational force from a real object can be considered to come from that object’s center.

Multiple Choice: 8

Short Answer: 1, 3

4.2 An Orbit Is One Body “Falling around” Another

Illustrate that orbits are a perpetual state of free fall.

Multiple Choice: 23

Short Answer: 6

Relate an object’s speed to its orbital path.

Multiple Choice: 19, 20, 21, 22

Show how Kepler’s laws are consistent with Newton’s universal law of gravitation.

Multiple Choice: 24

4.3 Tidal Forces Are Caused by Gravity

Use Newton’s universal law of gravitation to explain why objects of real physical size experience tidal forces.

Multiple Choice: 25, 26

Short Answer: 7

Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

Multiple Choice: 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37

Short Answer: 8, 9, 10, 11, 12, 13, 14, 19

4.4 Tides Affect Solid Bodies

Explain how tidal forces cause orbital locks.

Multiple Choice: 39, 40, 46, 50, 52

Short Answer: 15

Describe how tidal forces change the orbital and rotational periods of objects.

Multiple Choice: 38, 41, 42, 43, 51

Short Answer: 18

Compare and contrast how tidal forces affect approaching objects of different sizes.

Multiple Choice: 44, 45, 48, 49

Short Answer: 16, 17

Working It Out 4.1

Use proportional reasoning with Newton’s universal law of gravitation to explore how changing input parameters effects the resulting force.

Multiple Choice: 53, 54, 55, 56, 57, 58, 59

Short Answer: 20, 21, 22, 23

Calculate the gravitational acceleration on another planet or body.

Short Answer: 24, 25, 26

Working It Out 4.2

Calculate the circular and escape velocities from an object.

Multiple Choice: 60, 61, 62, 63, 64, 65

Short Answer: 27, 28

Working It Out 4.3

Calculate the mass of a central object using the orbital properties of a satellite.

Multiple Choice: 66, 67, 68, 69, 70

Short Answer: 29, 30

Working It Out 4.4

Compute the relative strength of tidal forces from different objects.

Multiple Choice: 71

MULTIPLE CHOICE

1. Two rocks (call them S and T) are released at the same time from the same height and start from rest. Rock S has 20 times the mass of rock T. Which rock will fall faster if the only forces involved are each rock’s mutual gravitational attraction with Earth?
   1. Rock S
   2. Rock T
   3. Both rocks will fall at the same rate.
   4. Not enough information is available to answer.

ANS: C         DIF: Easy              REF: Section 4.1

MSC: Applying

OBJ: Differentiate gravity, mass, and weight.

2. Which of the following properties of an astronaut changes when he or she is standing on the Moon, relative to when the astronaut is standing on Earth?
   1. weight
   2. mass
   3. inertia
   4. Nothing changes.

ANS: A         DIF: Easy              REF: Section 4.1

MSC: Applying

OBJ: Differentiate gravity, mass, and weight.

3. _________ hypothesized that planetary motions could be explained by a force arising from the attraction between the mass of the planet and the Sun that decreased with the square of the distance between them.
   1. Johannes Kepler
   2. Isaac Newton
   3. Tycho Brahe
   4. Nicolaus Copernicus
   5. Galileo Galilei

ANS: B         DIF: Easy              REF: Section 4.1

MSC: Remembering

OBJ: Define the bold-faced vocabulary terms within the chapter.

4. The force of gravity that an object has is directly proportional to its

ANS: C         DIF: Easy              REF: Section 4.1

MSC: Remembering

OBJ: Define the bold-faced vocabulary terms within the chapter.

5. If the distance between Earth and the Sun were cut in half, the gravitational force between these two objects would
   4. decrease by 4.
   5. decrease by 2.
   6. increase by 2.
   7. increase by 4.
   8. decrease by 8.

ANS: D         DIF: Easy              REF: Section 4.1

MSC: Applying

OBJ: Describe the behavior of the gravitational force using Newton’s universal law of gravitation.

6. According to the progression shown in the figure below, if the distance between two objects is increased to four times its original value, the gravitational force between the two objects would be _________ times its original value.
   1. 1/2
   2. 1/32
   3. 1/4
   4. 1/16
   5. 4

ANS: D         DIF: Easy              REF: Section 4.1

MSC: Understanding

OBJ: Describe the behavior of the gravitational force using Newton’s universal law of gravitation.

7. Newton’s gravitational constant is
   1. a force.
   2. a weight.
   3. a number.
   4. an acceleration.
   5. a mass.

ANS: C         DIF: Easy              REF: Section 4.1

MSC: Remembering

OBJ: Define the bold-faced vocabulary terms within the chapter.

8. The distance used in Newton’s law of gravitation is
   1. the distance between the closest faces of the two objects.
   2. the distance between the centers of the objects.
   3. the radius of the largest object.
   4. the radius of the smallest object.
   5. always the same.

ANS: B         DIF: Easy              REF: Section 4.1

MSC: Remembering

OBJ: Explain how gravitational force from a real object can be considered to come from that object’s center.

9. Gravity is
   1. the strongest force.
   2. a fundamental force.
   3. a mutually attractive force.
   4. a mutually repulsive force.
   5. both B and C

ANS: E          DIF: Easy              REF: Section 4.1

MSC: Remembering

OBJ: Define the bold-faced vocabulary terms within the chapter.

10. An astronaut who weighs 700 N on Earth is located in empty space, very far from any other objects. Approximately what is the mass of the astronaut?
    1. 0
    2. 7 kg
    3. 70 kg
    4. 700 kg
    5. 7000 kg

ANS: C         DIF: Medium        REF: Section 4.1

MSC: Applied

OBJ: Differentiate gravity, mass, and weight.

11. Two rocks (call them S and T) are a distance of 50 km from one another. Rock S has 20 times the mass of rock T. Considering only their mutual gravitational force, which rock will accelerate faster in response to gravity?
    1. rock S
    2. rock T
    3. Both rocks will have the same acceleration.
    4. Not enough information is available to answer.

ANS: B         DIF: Medium        REF: Section 4.1

MSC: Applying

OBJ: Use Newton’s universal law of gravitation to quantify the force of gravity between two objects in different physical situations.

12. In the absence of air friction, a 0.001-kg piece of paper and a 0.1-kg notebook are dropped from the same height and allowed to fall to the ground. How do their accelerations compare?
    1. The accelerations are the same.
    2. The notebook’s acceleration is 100 times faster than the paper’s acceleration.
    3. The notebook’s acceleration is 1,000 times faster than the paper’s acceleration.
    4. The paper’s acceleration is 100 times faster than the notebook’s acceleration.
    5. The paper’s acceleration is 1,000 times faster than the notebook’s acceleration.

ANS: A         DIF: Medium        REF: Section 4.1

MSC: Applying

OBJ: Differentiate gravity, mass, and weight.

13. According to the scales shown in the figure below, about how many times stronger is gravity on Earth than on the Moon?
    1. 20
    2. 3
    3. 2
    4. 6
    5. They are the same.

ANS: D         DIF: Medium        REF: Section 4.1

MSC: Applying

OBJ: Use Newton’s universal law of gravitation to quantify the force of gravity between two objects in different physical situations.

14. Which of the following is not true about orbits?
    1. Orbits are ellipses.
    2. Orbits can be circular.
    3. An orbit is the path of an object in free fall around another object.
    4. Orbits are always circular.
    5. Objects in orbits experience acceleration.

ANS: D         DIF: Easy              REF: Section 4.2

MSC: Remembering

OBJ: Define the bold-faced vocabulary terms within the chapter.

15. A centripetal force is
    1. a fundamental force.
    3. any force that points toward center of a circular path.
    4. tension force.
    5. magnetic force.

ANS: C         DIF: Easy              REF: Section 4.2

MSC: Remembering

OBJ: Define the bold-faced vocabulary terms within the chapter.

16. Uniform circular motion applies to which of the following orbits?
    1. elliptical
    2. hyperbolic
    3. circular
    4. parabolic

ANS: C         DIF: Easy              REF: Section 4.2

MSC: Remembering

OBJ: Define the bold-faced vocabulary terms within the chapter.

17. The term satellite in astronomy means
    1. a means of communication.
    2. a man-made object in orbit of Earth.
    3. the Moon.
    4. any low-mass object that is orbiting a more massive object.
    5. none of the above

ANS: D         DIF: Easy              REF: Section 4.2

MSC: Remembering

OBJ: Define the bold-faced vocabulary terms within the chapter.

18. Which of the following is a bound orbit?
    1. elliptical
    2. hyperbolic
    3. circular
    4. both A and C

ANS: D         DIF: Easy              REF: Section 4.2

MSC: Remembering

OBJ: Define the bold-faced vocabulary terms within the chapter.

19. For which of the following orbital velocities, V, is the orbit unbound?
    1. V = Escape velocity
    2. V = Circular velocity
    3. V > Escape velocity
    4. V < Escape velocity
    5. both A and C

ANS: E          DIF: Easy              REF: Section 4.2

MSC: Remembering

OBJ: Relate an object’s speed to its orbital path.

20. If an object is in orbit with an orbital speed less than the escape speed but greater than the circular orbit speed, what type of orbit is it?
    1. elliptical
    2. hyperbolic
    3. circular
    4. both A and C

ANS: A         DIF: Easy              REF: Section 4.2

MSC: Remembering

OBJ: Relate an object’s speed to its orbital path.

21. If an object is moving in a circular orbit at a constant speed, which of the following is false?
    1. Its acceleration is not zero.
    2. Its acceleration is zero.
    3. Its velocity is not zero.
    4. There is an unbalanced force acting on it.
    5. All the above statements are true.

ANS: B         DIF: Medium        REF: Section 4.2

MSC: Understanding

OBJ: Relate an object’s speed to its orbital path.

22. If we wanted to increase the Hubble Space Telescope’s altitude above Earth and keep it in a stable orbit, we also would need to
    1. increase its orbital speed.
    2. increase its weight.
    3. decrease its weight.
    4. decrease its orbital speed.
    5. increase its mass.

ANS: D         DIF: Medium        REF: Section 4.2

MSC: Applying

OBJ: Relate an object’s speed to its orbital path.

23. Astronauts orbiting Earth in the space shuttle experience so-called weightlessness in space because
    1. they are farther away from Earth.
    2. they eat less food while in orbit.
    3. the gravitational pull of the Moon counteracts Earth’s gravitational pull.
    4. they are in constant free fall around Earth.
    5. they are in space where there is no gravity.

ANS: D         DIF: Medium        REF: Section 4.2

MSC: Applying

OBJ: Illustrate that orbits are a perpetual state of free fall.

24. If you measured the orbital period of the Moon and the distance between Earth and the Moon, then you could calculate
    1. the mass of the Moon.
    2. the sum of the masses of Earth and the Moon.
    3. the average distance between Earth and the Sun.
    4. the radius of Earth.
    5. the radius of the Moon.

ANS: B         DIF: Medium        REF: Section 4.2

MSC: Understanding

OBJ: Show how Kepler’s Laws are consistent with Newton’s universal law of gravitation.

25. Why is Earth’s tidal bulge not perfectly aligned with the line connecting the centers of Earth and the Moon?
    1. friction
    2. Earth’s rotation
    3. gravity
    4. both A and B

ANS: D         DIF: Easy              REF: Section 4.3         MSC: Remembering

OBJ: Use Newton’s universal law of gravitation to explain why objects of real physical size experience tidal forces.

26. Tidal forces are caused by
    1. the weight of the water in the oceans on the ocean floor.
    2. the strength of the gravitation pull of the Moon on Earth.
    3. the difference between the weight of the water on the ocean floor at high and low tide.
    4. the difference between the strength of the gravitational pull of the Moon and Sun on either side of Earth.
    5. the strength of the gravitation pull of the Moon and the Sun on Earth.

ANS: D         DIF: Easy              REF: Section 4.3

MSC: Remembering

OBJ: Use Newton’s universal law of gravitation to explain why objects of real physical size experience tidal forces.

27. Lunar tides are approximately _________ solar tides.
    1. 2 times weaker than
    2. 2 times stronger than
    3. 200 times weaker than
    4. 200 times stronger than
    5. the same strength as

ANS: B         DIF: Easy              REF: Section 4.3

MSC: Remembering

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

28. According to the figure below, spring tides occur when the lunar and solar tides ________, resulting in _________ tides.
    1. add; above average
    2. partially cancel out; above average
    3. add; below average
    4. partially cancel out; below average
    5. completely cancel out; no

ANS: A         DIF: Easy              REF: Section 4.3

MSC: Understanding

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

29. According to the figure below, spring tides occur at which phases of the Moon?
    1. third quarter
    2. new and full
    3. first and third quarters
    4. full
    5. new

ANS: B         DIF: Easy              REF: Section 4.3

MSC: Understanding

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

30. According to the figure below, when the Moon is in between Earth and the Sun, _________ tides occur.
    1. spring
    2. no
    3. neap
    4. high
    5. low

ANS: A         DIF: Easy              REF: Section 4.3

MSC: Understanding

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

31. According to the figure below, when Earth is in between the Moon and the Sun, _________ tides occur.
    1. spring
    2. no
    3. neap
    4. high
    5. low

ANS: A         DIF: Easy              REF: Section 4.3

MSC: Understanding

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

32. According to the figure below, neap tides occur when the lunar and solar tides _________, resulting in _________ tides.
    1. add; above average
    2. partially cancel out; above average
    3. add; below average
    4. partially cancel out; below average
    5. completely cancel out; no

ANS: D         DIF: Easy              REF: Section 4.3

MSC: Understanding

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

33. According to the figure below, neap tides occur at which phases of the Moon?
    1. new and full
    2. third quarter
    3. first and third quarters
    4. full
    5. new

ANS: C         DIF: Easy              REF: Section 4.3

MSC: Understanding

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

34. When the Sun and Moon are separated by 90 degrees in the sky, _________ tides occur on Earth when the strength of the tides are _________ than normal.
    1. spring; lower
    2. spring; higher
    3. lunar; lower
    4. neap; lower
    5. neap; higher

ANS: D         DIF: Medium        REF: Section 4.3

MSC: Analyzing

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

35. According to the figure below, a high tide at a given location will occur about ____ time(s) a day.
    1. one
    2. three
    3. two
    4. four
    5. eight

ANS: C         DIF: Medium        REF: Section 4.3

MSC: Applying

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

36. According to the figure below, the approximate amount of time between two high tides at a given location is about
    1. 3 hours.
    2. 8 hours.
    3. 6 hours.
    4. 12 hours.
    5. 24 hours.

ANS: D         DIF: Medium        REF: Section 4.3

MSC: Applying

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

37. According to the figure below, the approximate amount of time between a high tide and a low tide at a given location is about
    1. 3 hours.
    2. 8 hours.
    3. 6 hours.
    4. 12 hours.
    5. 24 hours.

ANS: C         DIF: Medium        REF: Section 4.3

MSC: Applying

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

38. Because of tidal forces, which type of eclipse will become impossible first?
    1. partial lunar
    2. total lunar
    3. partial solar
    4. total solar
    5. annular solar

ANS: D         DIF: Difficult        REF: Section 4.4

MSC: Analyzing

OBJ: Describe how tidal forces change the orbital and rotational periods of objects.

39. In the figure below, the person on the Moon is standing at the same location on the Moon as the Moon rotates around its own axis. Based on this, we see that the Moon’s rotational period about its own axis is equal to
    1. Earth’s rotational period.
    2. half Earth’s rotational period.
    3. the Moon’s orbital period.
    4. half the Moon’s orbital period.
    5. Earth’s orbital period around the Sun.

ANS: C         DIF: Easy              REF: Section 4.4

MSC: Understanding

OBJ: Explain how tidal forces cause orbital locks.

40. The Moon always keeps the same face toward Earth because of
    1. tidal locking.
    2. tidal forces from the Sun.
    3. tidal forces from Earth.
    4. tidal forces from Earth and the Sun.
    5. all sides of the moon face Earth at one time or another.

ANS: A         DIF: Easy              REF: Section 4.4

MSC: Remembering

OBJ: Explain how tidal forces cause orbital locks.

41. The distance between Earth and the Moon
    1. will never change.
    2. is slowly decreasing.
    3. is slowly increasing.
    4. will increase or decrease depending on future changes in the tides on the Moon due to Earth.
    5. will increase or decrease depending on future changes in the tides on Earth due to the Moon.

ANS: C         DIF: Easy              REF: Section 4.4

MSC: Remembering

OBJ: Describe how tidal forces change the orbital and rotational periods of objects.

42. Earth’s rotation rate is slowing because of
    1. radioactive decays in its core.
    2. relativistic effects of gravity.
    3. tidal forces from the Moon.
    4. the gravitational force of the Sun.
    5. gravitational drag from dark matter.

ANS: C         DIF: Easy              REF: Section 4.4

MSC: Applying

OBJ: Describe how tidal forces change the orbital and rotational periods of objects.

43. The distance between Earth and the Moon is increasing because of
    1. the expansion of the universe.
    2. the expansion of the Solar System.
    3. tidal forces from the Moon.
    4. tidal forces from the Moon and Sun.
    5. dark energy.

ANS: C         DIF: Easy              REF: Section 4.4

MSC: Applying

OBJ: Describe how tidal forces change the orbital and rotational periods of objects.

44. Which one of the statements below about a planet’s Roche limit is false?
    2. The Roche limit is about 2.5 times the radius of gaseous planets.
    3. Objects orbiting closer to a planet than the Roche limit are likely to be ripped apart by tidal forces.
    4. The Roche limit is where tidal forces from an orbiting object are equal to its internal self-gravity.
    5. Orbiting objects beyond the Roche limit from the planet do not get ripped apart by tidal forces.
    6. The ring systems around giant planets are located beyond the Roche limit.

ANS: E          DIF: Easy              REF: Section 4.4

MSC: Understanding

OBJ: Compare and contrast how tidal forces affect approaching objects of different sizes.

45. Tidal forces can affect
    5. all of the above

ANS: E          DIF: Easy              REF: Section 4.4

MSC: Remembering

OBJ: Compare and contrast how tidal forces affect approaching objects of different sizes.

46. _________ may have been instrumental in shaping the interface between Earth’s land and oceans where the chemistry needed to develop life may have occurred.
    1. Meteor showers
    2. Collisions with comets
    3. Earth’s Moon
    4. Tectonic activity
    5. A collision with a Mars-sized object

ANS: C         DIF: Medium        REF: Section 4.4

MSC: Remembering

OBJ: Explain how tidal forces cause orbital locks.

47. The Roche limit
    1. is the distance at which a planet’s tidal forces become equal to self-gravity of an object.
    2. is The limit on the amount of mass an object can have in orbit.
    3. is the smallest orbit possible around a planet.
    4. only applies to the giant planets.
    5. only applies to stars.

ANS: A         DIF: Medium        REF: Section 4.4

MSC: Remembering

OBJ: Define the bold-faced vocabulary terms within the chapter.

48. When two galaxies collide long streams of stars can be observed. These “tails” are caused by
    2. magnetic forces.
    3. tidal forces.
    4. Roche forces.
    5. dark energy.

ANS: C         DIF: Medium        REF: Section 4.4

MSC: Remembering

OBJ: Compare and contrast how tidal forces affect approaching objects of different sizes.

49. Which of the statements below are true about the Roche limit of a giant planet?
    1. It is about equal to the radius of the planet.
    2. It is the closest to the planet that moons normally are found.
    3. It is the closest to the planet that rings will be found.
    4. It is the farthest from the planet that moons normally are found.
    5. Because they have no solid surfaces, giant planets do not have a Roche limit.

ANS: B         DIF: Medium        REF: Section 4.4

MSC: Understanding

OBJ: Compare and contrast how tidal forces affect approaching objects of different sizes.

50. The moon keeps the same hemisphere facing Earth because the _________ is equal to the _________.
    1. rotational period of Earth; orbital period of the Moon around Earth
    2. orbital period of Earth; orbital period of the Moon around Earth
    3. orbital period of the Moon around Earth; rotational period of Earth
    4. rotational period of the Moon; orbital period of the Moon around Earth
    5. rotational period of Earth; orbital period of Earth

ANS: D         DIF: Medium        REF: Section 4.4

MSC: Remembering

OBJ: Explain how tidal forces cause orbital locks.

51. Because of the tidal force between Earth and the Moon,
    1. Earth’s rotation rate is decreasing.
    2. the Moon’s distance from Earth is increasing.
    3. the Moon’s orbital period is increasing.
    4. the Moon’s rotational period is increasing.
    5. all of the above are true.

ANS: E          DIF: Difficult        REF: Section 4.4

MSC: Remembering

OBJ: Describe how tidal forces change the orbital and rotational periods of objects.

52. Because of tidal forces, for every _________ time(s) it rotates on its axis, Mercury revolves around the Sun _________ time(s).
    1. 1; 1
    2. 2; 3
    3. 3; 2
    4. 10; 1
    5. 20; 1

ANS: C         DIF: Difficult        REF: Section 4.4

MSC: Remembering

OBJ: Explain how tidal forces cause orbital locks.

53. Suppose you are suddenly transported to a planet with 1/4 the mass of Earth but the same radius as Earth. Your weight would _________ by a factor of _________.
    1. increase; 4
    2. increase; 16
    3. decrease; 4
    4. decrease; 16
    5. increase; 2

ANS: C         DIF: Medium        REF: Working It Out 4.1

MSC: Applying

OBJ: Use proportional reasoning with Newton’s universal law of gravitation to explore how changing input parameters effects the resulting force.

54. Suppose you are suddenly transported to a planet that had 1/4 the radius of Earth but the same mass as Earth. Your weight would _________ by a factor of _________.
    1. increase; 4
    2. increase; 16
    3. decrease; 4
    4. decrease; 16
    5. decrease; 8

ANS: B         DIF: Medium        REF: Working It Out 4.1

MSC: Applying

OBJ: Use proportional reasoning with Newton’s universal law of gravitation to explore how changing input parameters effects the resulting force.

55. If you weighed 150 lb on Earth, what would you weigh on Mars? For reference, Mars has a mass that is 0.1 times Earth’s mass and Mars has a radius that is 0.5 times Earth’s radius.
    1. 30 lb
    2. 110 lb
    3. 75 lb
    4. 60 lb
    5. 15 lb

ANS: D         DIF: Difficult        REF: Working It Out 4.1

MSC: Applying

OBJ: Use proportional reasoning with Newton’s universal law of gravitation to explore how changing input parameters effects the resulting force.

56. If you weighed 100 lb on Earth, what would you weigh at the upper atmosphere of Jupiter? For reference, Jupiter has a mass that is about 300 times Earth’s mass and a radius that is 10 times Earth’s radius.
    1. 10,000 lb
    2. 3,000 lb
    3. 1,000 lb
    4. 300 lb
    5. 30 lb

ANS: D         DIF: Difficult        REF: Working It Out 4.1

MSC: Applying

OBJ: Use proportional reasoning with Newton’s universal law of gravitation to explore how changing input parameters effects the resulting force.

57. The force of gravity between Earth and the Sun is _________ the force of gravity between Earth and the Moon. For reference, the average distance between Earth and the Moon is 0.003 astronomical unit (AU), the mass of the Moon is 7 × 10²² kg, and the mass of the Sun is 2 × 10³⁰
    1. 86,000 times larger than
    2. 260 times larger than
    3. 140 times smaller than
    4. 6,400 times smaller than
    5. the same as

ANS: B         DIF: Difficult        REF: Working It Out 4.1

MSC: Applying

OBJ: Use proportional reasoning with Newton’s universal law of gravitation to explore how changing input parameters effects the resulting force.

58. The force of gravity between Saturn and the Sun is _________ the force of gravity between Earth and the Sun. For reference, Saturn is approximately 100 times more massive than Earth, and the semimajor axis of Saturn’s orbit is 10 AU.
    1. 10 times smaller than
    2. 1,000 times larger than
    3. 1,000 times smaller than
    4. 100 times larger than
    5. approximately equal to

ANS: E          DIF: Difficult        REF: Working It Out 4.1

MSC: Applying

OBJ: Use proportional reasoning with Newton’s universal law of gravitation to explore how changing input parameters effects the resulting force.

59. Mercury orbits the Sun with an average distance of 0.4 AU, and its mass is 0.06 times that of Earth. The gravitational force that the Sun exerts on Mercury is _______________ times the force of gravity that the Sun exerts on Earth.
    1. 20
    2. 6
    3. 4
    4. 4
    5. 1

ANS: D         DIF: Difficult        REF: Working It Out 4.1

MSC: Applying

OBJ: Use proportional reasoning with Newton’s universal law of gravitation to explore how changing input parameters effects the resulting force.

60. If you have two moons that have the same radius, but Moon A is denser and has 2 times the mass of Moon B, how do their escape velocities compare?
    1. Moon A has an escape velocity that is 1.4 times larger than Moon B.
    2. Moon A has an escape velocity that is 1.4 times smaller than Moon B.
    3. Moon A has an escape velocity that is 2 times smaller than Moon B.
    4. Moon A has an escape velocity that is 2 times larger than Moon B.
    5. Because gravity affects all masses the same, the escape velocities are the same.

ANS: A         DIF: Easy              REF: Working It Out 4.2

MSC: Applying

OBJ: Calculate the circular and escape velocities from an object.

61. The Hubble Space Telescope orbits at an altitude of 600 km above Earth’s surface. Assuming it is in a stable circular orbit, what is its velocity? For reference, Earth’s radius is 6,400 km and Earth’s mass is 6 × 10²⁴
    1. 240,000 m/s
    2. 7,500 m/s
    3. 51,000 m/s
    4. 64,000 m/s
    5. You also must know the mass of the Hubble Space Telescope to determine its speed.

ANS: B         DIF: Difficult        REF: Working It Out 4.2

MSC: Applying

OBJ: Calculate the circular and escape velocities from an object.

62. How fast is the Moon moving as it orbits Earth? For reference, the Moon’s orbit is approximately circular with a radius equal to 400,000 km, and the Moon’s orbital period is 27 days.
    1. 1 km/s
    2. 10 km/s
    3. 50 km/
    4. 100 km/s
    5. 500 km/s

ANS: A         DIF: Difficult        REF: Working It Out 4.2

MSC: Applying

OBJ: Calculate the circular and escape velocities from an object.

63. What is the escape velocity from Mars if its mass is 6 × 10²³ kg and its radius is 3,400 km?
    1. 2,400 m/s
    2. 4,900 m/s
    3. 8,600 km/s
    4. 12,000 m/s
    5. 25,000 km/s

ANS: B         DIF: Difficult        REF: Working It Out 4.2

MSC: Applying

OBJ: Calculate the circular and escape velocities from an object.

64. What is the escape velocity from a large asteroid if its mass is 6 × 10²¹ kg and its radius is 2,400 km?
    1. 98 km/s
    2. 210 km/s
    3. 340 m/s
    4. 580 m/s
    5. 12,400 m/s

ANS: D         DIF: Difficult        REF: Working It Out 4.2

MSC: Applying

OBJ: Calculate the circular and escape velocities from an object.

65. If a satellite sent to Mars is designed to return a rock sample to Earth, how fast must the satellite be launched from its surface in order to escape Mars’s gravity? For reference, Mars has a mass of 6 × 10²³ kg and a radius of 3,400 km.
    1. 100 m/s
    2. 5,000 m/s
    3. 20 m/s
    4. 20,000 m/s
    5. You must know the mass of the satellite to determine the answer.

ANS: B         DIF: Difficult        REF: Working It Out 4.2

MSC: Applying

OBJ: Calculate the circular and escape velocities from an object.

66. If you found an exoplanet whose mass was the same as Jupiter’s, but the planet orbited its star with a period of 2 years and a semimajor axis of 1 AU, what would be the mass of its star? For reference, Jupiter has a semimajor axis of 5.4 AU and an orbital period of 12 years.
    1. 25 M_Sun
    2. 5 M_Sun
    3. 0 M_Sun
    4. 5 M_Sun
    5. Not enough information is available to answer.

ANS: A         DIF: Difficult        REF: Working It Out 4.3

MSC: Applying

OBJ: Calculate the mass of a central object using the orbital properties of a satellite.

67. Titan, the largest moon of Saturn, has an orbital period of 16 days and a semimajor axis of 1.2 × 10⁹ Based on this information, what is Saturn’s mass? For reference, Earth’s mass is 6 × 10²⁴ kg.
    1. 290 M_Earth
    2. 130 M_Earth
    3. 90 M_Earth
    4. 40 M_Earth
    5. 4 M_Earth

ANS: C         DIF: Difficult        REF: Working It Out 4.3

MSC: Applying

OBJ: Calculate the mass of a central object using the orbital properties of a satellite.

68. You find a moon orbiting a planet. The moon has a period of 10 days, and the average distance between the moon and planet is 10⁶ What is the planet’s mass? Note that the mass of Jupiter is 1.9 × 10²⁷ kg.
    1. 1 M_Jupiter
    2. 4 M_Jupiter
    3. 1 M_Jupiter
    4. 4 M_Jupiter
    5. 10 M_Jupiter

ANS: B         DIF: Difficult        REF: Working It Out 4.3

MSC: Applying

OBJ: Calculate the mass of a central object using the orbital properties of a satellite.

69. If you discovered a planet orbiting another star, and the planet had an orbital period of 2 years and a semimajor axis of 2 AU, what would be the mass of its parent star? You can assume the planet’s mass is much less than the star’s mass.
    1. 25 M_Sun
    2. 5 M_Sun
    3. 0 M_Sun
    4. 25 M_Sun
    5. 0 M_Sun

ANS: E          DIF: Difficult        REF: Working It Out 4.3

MSC: Applying

OBJ: Calculate the mass of a central object using the orbital properties of a satellite.

70. Assume that a planet just like Earth orbits the bright star named Sirius. If this Earth-like planet orbits with a semimajor axis of 1 AU and an orbital period of 7 months, what is the mass of Sirius?
    1. 3 M_Sun
    2. 12 M_Sun
    3. 8 M_Sun
    4. 5 M_Sun
    5. 17 M_Sun

ANS: A         DIF: Difficult        REF: Working It Out 4.3

MSC: Applying

OBJ: Calculate the mass of a central object using the orbital properties of a satellite.

71. If you doubled the distance the Moon is from Earth, by what fraction does the strength of the tidal force change?
    1. 2
    2. 1/2
    3. 1/4
    4. 1/8
    5. 1/16

ANS: D         DIF: Difficult        REF: Working It Out 4.4

MSC: Remembering

OBJ: Compute the relative strength of tidal forces from different objects.

SHORT ANSWER

1. Is there a difference in your weight when measured on top of a mountain 1,000 meters above sea level and when measured in a classroom 10 meters above sea level?

ANS: Yes, you weigh less on the mountaintop because the distance between you and the center of Earth (R) is larger. However, the difference in weight is so small that it isn’t noticeable, because Earth is so large (R = 6,370 km) compared to the 1-km difference in these altitudes.

DIF: Easy  REF: Section 4.1  MSC: Applying

OBJ: Explain how gravitational force from a real object can be considered to come from that object’s center.

2. Newton’s law of gravity says that gravity is a mutually attractive force. Explain the following observation. A small object is dropped on Earth and we see it fall toward Earth. However, we do not observe Earth moving toward the object.

ANS: Both Earth and the object experience a force of the same magnitude. However, Earth is so much more massive that the acceleration it experiences is negligible, and therefore not observable. However, the small object’s mass is very small in comparison, so we observe it to accelerate toward Earth.   DIF: Easy  REF: Section 4.1   MSC: Understanding

OBJ: Describe the behavior of the gravitational force using Newton’s universal law of gravitation.

3. Explain why the gravitational force an object experiences from Earth can be considered to come from the center of Earth.

ANS: Each part of Earth pulls on the object by a different amount, which depends on the distance from the object to that part of Earth. The sum of all these forces, or the net force, is the same as if all the mass were located at the center of Earth.

DIF: Medium  REF: Section 4.1   MSC: Understanding

OBJ: Explain how gravitational force from a real object can be considered to come from that object’s center.

4. Explain what the terms circular velocity and escape velocity Give the formula for each and explain what each mathematical symbol represents.

ANS: The circular velocity is the velocity that an object needs in order to maintain a stable orbit around an object of mass M at a distance r from it, and the circular velocity is equal to v_circ = (GM/r)^1/2. The escape velocity is the minimum velocity an object would have to be given if it were to escape the gravity of the object it orbits, whose mass is M. If the object is a distance R from the object it orbits, then the escape velocity is equal to v_esc = (2GM/R)^1/2.

DIF: Easy  REF: Section 4.2

MSC: Remembering

OBJ: Define the bold-faced vocabulary terms within the chapter.

5. Explain the difference between a bound orbit and an unbound orbit.

ANS: A bound orbit is a circular or elliptical orbit; the orbital speeds are less than the escape speed. An unbound orbit is parabolic or hyperbolic, where the orbital velocity is equal to or greater than the escape speed. In a bound orbit, the object continues to go around the central object. In an unbound orbit the object makes one pass by the central object and then continues out to infinity, never coming back.

DIF: Easy  REF: Section 4.2

MSC: Understanding

OBJ: Define the bold-faced vocabulary terms within the chapter.

6. Explain the difference between being weightless and being in free fall.

ANS: The only way to be truly weightless is to have no mass. If you stand on a scale in free-fall, both you and the scale are falling at the same rate, therefore you exert no force on the scale hence the term “weightlessness”.

DIF: Medium  REF: Section 4.2

MSC: Understanding

OBJ: Illustrate that orbits are a perpetual state of free fall.

7. Explain the origin of tidal forces on Earth due to the Moon.

ANS: Because Earth is a real object with size, the gravitational force between the Moon and the various parts of Earth varies because the distance to each part of Earth is different. The side of Earth closest to the Moon experiences the greatest force, while the far side of Earth from the Moon experiences the weakest force. The differences in the Moon’s attraction to parts of Earth are called the tidal forces.

DIF: Easy  REF: Section 4.3   MSC: Understanding

OBJ: Use Newton’s universal law of gravitation to explain why objects of real physical size experience tidal forces.

8. Do tidal forces only affect the water on Earth?

ANS: No, tidal forces also distort Earth’s crust. However, since the crust is solid its response to the tidal force is less.

DIF: Easy  REF: Section 4.3  MSC: Analyzing

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

9. Why are there high and low tides each day instead of having the same tide all day during a given phase of the Moon?

ANS: There are high and low tides each day because Earth rotates faster than the tidal bulge.

DIF: Easy  REF: Section 4.3   MSC: Understanding

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

10. If Earth did not have a moon, would we still have tides because of the Sun.? If so, explain how they might be different, or why they might remain unchanged.

ANS: If Earth did not have a moon, Earth would still have tides because of the Sun. However, the variation between low and high tide would be less. There would be no spring and neap tides. High and low tide would always be the same “height.”

DIF: Medium  REF: Section 4.3  MSC: Analyzing

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

11. Consider the figure below, which illustrates the tidal bulge on Earth’s oceans due to the Moon and four people at different longitudes on Earth from the point of view of an observer looking down on the North Pole of Earth. If you were to arrive at the beach and find that the Moon was visible in the western half of the sky, then is the tide most likely to be coming in and the water level rising; or is the tide going out and the water level going down? Explain the rationale for your answer.

ANS: If the Moon is in the western half of the sky, then you would be located somewhere between the person shown on the right, for whom the Moon is just west of the meridian, and the person shown on the top, for whom the Moon has just set. Therefore the tide would be going out and the water level going would be going down.   DIF: Medium  REF: Section 4.3  MSC: Applying

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

12. Which is larger, the tidal force on Earth due to the Moon or the tidal force on Earth due to the Sun, and by approximately how much? Explain conceptually why this is possible given that the gravitational force of the Sun on Earth is 200 times larger than the gravitational force of the Moon on Earth.

ANS: Tides are caused by the difference between the gravity affecting one side of an object and the other. The tidal force on Earth due to another body of mass M is F = 2GMM_ER_E/d³, where d is the distance between the body and Earth. When you do the calculation, the tidal force of the Moon on Earth is about two times that of the Sun. Even though the mass of the moon is smaller than the Sun, its much closer distance makes the tidal force larger.

DIF: Medium  REF: Section 4.3   MSC: Understanding

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

13. Consider the figure below, which illustrates the tidal bulge on Earth’s oceans due to the Moon and Sun from an observer looking down on the North Pole of Earth. At what phase of the Moon will the lowest tides of the year occur? Explain the rationale for your answer either in words or with a sketch.

ANS: The lowest tides of the year are called neap tides, and they occur when the Sun’s gravitational pull is at right angles to the Moon’s gravitational pull. The neap tides are shown in part (b) of the figure, and they occur for the first-quarter moon (top of the panel) and the third-quarter moon (bottom of the panel).

DIF: Medium  REF: Section 4.3   MSC: Understanding

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

14. Show that the tidal force on Earth from the Moon is approximately two times the tidal force on Earth from the Sun. For reference, the Moon’s mass is 7.3 ×10²² kg, the Sun’s mass is 2 × 10³⁰ kg, the Earth−Moon distance is 3.8 × 10⁵ km, and the Earth−Sun distance is 1.5 × 10⁸

ANS: The tidal force on Earth due to a body of mass M at a distance d is given by F = 2GM M_E R_E/d³. Thus the ratio of the tidal force on Earth due to the Moon compared to the Sun is F_M/F_S = (2GM_M M_E R_E/d_EM³)/(2GM_S M_E R_E/d_ES³) = (M_M/M_S) × (d_ES/d_EM)³ = (7.3 × 10 ²² kg/2 × 10³⁰ kg) × ( 1.5 × 10⁸ km/3.8 × 10⁵ km)³ = 2.2.

DIF: Difficult  REF: Section 4.3   MSC: Applying

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

15. Explain why the Moon rotates in the same amount of time as it takes to orbit once around Earth.

ANS: The Moon’s tidal locking is caused by tidal friction that results in the side of the Moon that is heavier always facing Earth.

DIF: Easy  REF: Section 4.4   MSC: Understanding

OBJ: Explain how tidal forces cause orbital locks.

16. Explain what the Roche limit is and how it is related to rings around giant planets.

ANS: The Roche limit is the distance from a large planet at which the tidal forces on an object, such as a moon, are equal to the self-gravity that holds the object together. It is equal to approximately 2.5 times the radius of a large planet. Outside this distance moons can exist, but inside this distance they are likely to be torn apart by tidal forces and, in doing so, create planetary rings.

DIF: Easy  REF: Section 4.4   MSC: Understanding

OBJ: Compare and contrast how tidal forces affect approaching objects of different sizes.

17. Explain why Saturn’s rings do not clump together to form a moon.

ANS: Saturn’s rings orbit within the Roche limit. Inside of the Roche limit the tidal forces are stronger than the self-gravity that would hold a moon together.

DIF: Easy  REF: Section 4.4  MSC: Understanding

OBJ: Compare and contrast how tidal forces affect approaching objects of different sizes.

18. Earth’s tidal bulge “leads” the Moon in its orbit. Does this have any effect on the Moon?

ANS: Yes, the near and far side of Earth’s tidal bulge each pulls on the Moon. However, since the distances to each side of the bulge differ the magnitude of the force from the near side of the bulge is larger than that from the far side. Additionally, since the near side of the bulge is ahead of the Moon, there is a component of the net force in the direction of the Moon’s orbital motion. This causes the Moon to accelerate, increasing the size of its orbit over time.

DIF: Medium  REF: Section 4.4   MSC: Remembering

OBJ: Describe how tidal forces change the orbital and rotational periods of objects.

19. Imagine a planet in a very eccentric elliptical orbit around a Star. This planet has no moons, but it has oceans. This planet’s orbit is not tidally locked. Are there tides? If so, explain how they would behave.

ANS: The planet would have the equivalent of solar tides. The planet will experience about 2 high and 2 low tides each time the planet rotates once around its own axis. When the planet is near the star, the difference in height between low and high tide would be the most extreme. When the planet is furthest from the star the difference in height between high and low tide would be less extreme.

DIF: Difficult  REF: Section 4.3   MSC: Analyzing

OBJ: Characterize how tidal forces from the Moon and Sun cause the rise and fall of Earth’s ocean tides.

20. Show that the gravitational pull on Earth from the Sun is about 200 times the gravitational pull on Earth from the Moon. For reference, Moon’s mass is 7.3 × 10 ²² kg, the Sun’s mass is 2 × 10³⁰ kg, the Earth−Moon distance is 3.8 × 10⁵ km, and the Earth−Sun distance is 1.5 × 10⁸

ANS: F_grav = GM₁M₂/r²; thus F_Sun/F_Moon = (GM_SM_E/d_ES²)/ (GM_MM_E/d_EM²) = (M_Sun/M_Moon) × (d_EM/d_ES)² = (2 × 10³⁰ kg/7.3 × 10²² kg ) × (3.8 × 10⁵ km/1.5 × 10⁸ km)² = 176, which is approximately 200.

DIF: Medium  REF: Working It Out 4.1

MSC: Applying

OBJ: Use proportional reasoning with Newton’s universal law of gravitation to explore how changing input parameters effects the resulting force.

21. How much stronger is the gravitational force of the Sun on Earth compared to the gravitational force of the Sun on Pluto? Note that Pluto’s semimajor axis is 40, AU, and Pluto’s mass is 0.002 times the mass of Earth.

ANS: For a planet, the force of gravity between it and the Sun is F = GM_SunM/d²; thus F_P/F_E = (GM_SunM_P/d_P²)/ (GM_SunM_E/d_E²) = (M_P/M_E) × (d_E/d_P)² = 0.002 × (1/40)² = 0.002/40² = 1.3 × 10⁶. Therefore, the force of gravity between the Sun and Pluto is about a million times less than that between the Sun and Earth.

DIF: Medium  REF: Working It Out 4.1

MSC: Applying

OBJ: Use proportional reasoning with Newton’s universal law of gravitation to explore how changing input parameters effects the resulting force.

22. How does the force of gravity between the Sun and Mercury compare to the gravitational force between the Sun and Earth? Note that the semimajor axis of Mercury’s orbit is 0.4 AU, and Mercury’s mass is 0.06 times the mass of Earth.

ANS: For a planet, the force of gravity between it and the Sun is F = GM_SunM/d²; thus F_M/F_E = (GM_SunM_M/d_M²)/ (GM_SunM_E/d_E²) = (M_M/M_E) × (d_E/d_M)² = 0.06 × (1/0.4)² = 0.06/0.4² = 0.38. Therefore the force of gravity between the Sun and Mercury is only 38% that between the Sun and Earth.

DIF: Medium  REF: Working It Out 4.1   MSC: Applying

OBJ: Use proportional reasoning with Newton’s universal law of gravitation to explore how changing input parameters effects the resulting force.

23. How does the force of gravity between the Sun and Jupiter compare to the gravitational force between the Sun and Earth? Note that the semimajor axis of Jupiter’s orbit is 5.2 AU, and Jupiter’s mass is 320 times the mass of Earth.

ANS: For a planet, the force of gravity between it and the Sun is F = GM_SunM/d²; thus F_J/F_E = (GM_SunM_J/d_J²)/ (GM_SunM_E/d_E²) = (M_J/M_E) × (d_E/d_J)² = 320 × (1/5.2)² = 320/5.2² = 12. Therefore, the force of gravity between the Sun and Jupiter is 12 times larger than that between the Sun and Earth.

DIF: Medium  REF: Working It Out 4.1   MSC: Applying

OBJ: Use proportional reasoning with Newton’s universal law of gravitation to explore how changing input parameters effects the resulting force.

24. How would the acceleration due to gravity on a planet that is 16 times as massive as Earth and 4 times its radius compared to the acceleration of gravity on Earth?

ANS: For that planet, the acceleration due to gravity is g_planet = GM/R² = G(16M)/(4R)² = (16GM)/(16R²) = GM/R² = g. Therefore, the acceleration due to gravity would be the same for that planet as for Earth.

DIF: Difficult  REF: Working It Out 4.1

MSC: Applying

OBJ: Calculate the gravitational acceleration on another planet or body.

25. Saturn has 95 times the mass of Earth, and its atmosphere extends outward 9.5 times Earth’s radius. How does the acceleration due to gravity at the edge of Saturn’s atmosphere compare to that on Earth?

ANS: g = GM/R²; thus g_S/g_E = (GM_S/R_S²)/(GM_E/R_E²) = (M_S/M_E) × (R_E/R_S)² = 95 × (1/9.5)² = 1.05.

DIF: Difficult  REF: Working It Out 4.1

MSC: Applying

OBJ: Calculate the gravitational acceleration on another planet or body.

26. Mars has about one-tenth the mass of Earth and half Earth’s radius. How does the acceleration of gravity on Mars compare to that on Earth?

ANS: g = GM/R²; thus g_M/g_E = (GMM_M/R_M²)/(GMM_E/R_E²) = (M_M/M_E) × (R_E/R_M)² = (1/10) × 2² = 0.4.

DIF: Difficult  REF: Working It Out 4.1

MSC: Applying

OBJ: Calculate the gravitational acceleration on another planet or body.

27. What is the velocity one would need to give a satellite in order for it to escape from the Solar System (meaning escape the Sun’s gravity) if it was launched from Earth at a distance of 1 AU from the Sun? Give your answer in units of m/s.

ANS: The escape velocity from the Sun would be  where R = 1AU = 1.5 × 10¹¹m and the mass of the sun is M = 2.0 × 10³⁰ kg.

DIF: Medium  REF: Working It Out 4.2

MSC: Applying

OBJ: Calculate the circular and escape velocities from an object.

28. The International Space Station orbits at an altitude of 400 km above Earth’s surface. Assuming it is in a stable circular orbit, what is its velocity? For reference, Earth’s radius is 6,400 km, and Earth’s mass is 6 × 10²⁴ kg and G =7 × 10⁻¹¹ N m²/kg².

ANS: Recall that 1N = 1 kg m/s². Thus v_circ = (GM/r)^1/2 = (6.7 × 10⁻¹¹ m³/(kg s²) × 6 × 10²⁴ kg/[(6400 + 400) × 10³ m)]^1/2 = 7,700 m/s = 7.7 km/s.

DIF: Difficult  REF: Working It Out 4.2

MSC: Applying

OBJ: Calculate the circular and escape velocities from an object.

29. What two pieces of information would you need to obtain about one of the moons of the planet Jupiter in order to measure the mass of Jupiter? What formulae would you use to determine the mass?

ANS: You would need to know the semimajor axis of the moon’s orbit and the moon’s orbital period. You would use Newton’s version of Kepler’s third law to calculate the mass: M = 4π²A³/GP².

DIF: Medium  REF: Working It Out 4.3   MSC: Applying

OBJ: Calculate the mass of a central object using the orbital properties of a satellite.

30. You discover a moon orbiting a planet. The moon has an orbital period of three weeks, and the average distance between the moon and planet is 1.2 × 10⁶ What is the planet’s mass? Compare its mass to that of Jupiter, which is 1.9 × 10²⁷ kg.

ANS: Use the equation M = 4π²A³/GP², where G = 6.67 × 10⁻¹¹ Nm²/kg². First convert the units of G, P, and A so they contain combinations of only the units of kg, m, and s.

G = 6.67 × 10⁻¹¹ m³/kg s²

P = 3 weeks × (7 day/1 week) × (24 hr/1 day) × (3,600 sec/1 hr) = 1.81 × 10⁶ s

A = 1.2 × 10⁹ m

M = 4σ²(1.2 × 10⁹ m)³/(6.67 × 10⁻¹¹ m³/kg s² × (1.81 × 10⁶s)²) = 3.1 × 10²⁶ kg

M = 3.1 × 10²⁶ kg × (M_Jupiter/1.9 × 10²⁷ kg) = 0.16 M_Jupiter

DIF: Difficult  REF: Working It Out 4.3   MSC: Applying

OBJ: Calculate the mass of a central object using the orbital properties of a satellite.