=1.0 N kg−1
Worked example 3.4B.
A satellite of mass 1500 kg is in orbit around the Earth at an altitude of 500 km. It then uses its booster rockets to move to a higher orbit of altitude 3000 km. The Earth has a mass of 6.0 ×1024kg and a radius of 6.4 ×106m. The gravitational force on a 1500 kg mass in the Earth’s gravitational field is shown in the graph below.
500 500 F m
5.46 × 108 0.5 × 500
1 2
(a)
6.4 × 106 m 500 km
3000 km
Gravitational force on satellite (× 103 N)
7.0 6.0
15
10
5
0
9.0
8.0 10.0
(b) Students of mathematics will appreciate
that the area under the force–distance graph can also be determined by using calculus:
∆E as a body moves between altitudes A and B
=area = F.dR= dR
= −GMmR
]
B
A
[
GMm R2
B
∫
A B∫
APhysics file
a Calculate the speed of the satellite in its lower orbit.
b How fast does the satellite travel when in its higher orbit?
c How much gravitational potential energy does the satellite gain as it moves to the higher orbit?
d Calculate the change in kinetic energy experienced by the satellite as it changes orbit.
e How much energy does it take to boost the satellite into this higher orbit?
Solution
Lower radius =6.4 ×106m +500 km
=6.4 ×106m +5.0 ×105m
=6.9 ×106m
Higher radius =6.4 ×106m +3000 km
=6.4 ×106m +3.0 ×106m
=9.4 ×106m
a To determine the orbital speed of the satellite in the lower orbit:
a=
=
⇒v=
=
=7.6 ×103m s−1
b The speed of the satellite in the higher orbit is:
v=
=
=6.5 ×103m s−1
The satellite travels at 7.6 km s−1in the lower altitude orbit and at 6.5 km s−1in the higher orbit. As with all orbiting objects, it requires a greater speed to maintain a stable orbit when it travels through a region of stronger gravitational field.
c As the satellite moves to a higher altitude, it will gain potential energy.
This change in potential energy can be found by estimating the area under the graph between the orbital radii. From the graph, the increase in potential energy is:
120 squares ×(1 ×103) ×(0.2 ×106) J =2.4 ×1010J.
d It is not appropriate to use the graph to determine the change in kinetic energy of the satellite since it is not moving freely through the field. It is being powered by rockets to reach the higher altitude orbit. However, the kinetic energies can be found since the speeds in the different orbits are known.
Kinetic energy in lower orbit is:
Ek= mv2
=0.5 ×1500 ×(7.6 ×103)2
=4.3 ×1010J
1 2
6.67 × 10−11 × 6.0 × 1024 9.4 × 106
GM R
6.67 × 10−11 × 6.0 × 1024 6.9 × 106
GM R GM
R2 v2 R
e In its higher orbit , the satellite has 2.4 ×1010J more potential energy, but 1.4 ×1010J less kinetic energy. So overall, it has 1.0 ×1010J more energy. This is the amount of energy that the rockets provided to boost it from the lower orbit into the higher orbit.
Astronauts in orbit seem to ‘float’ around as if there were no gravity, yet they are in orbit only a few hundred kilo-metres above the Earth’s surface where the gravitational forces are still strong. At an altitude of 400 km (a typical shuttle orbit), the gravitational field strength is 8.7 N kg−1, almost as strong as the field strength at the Earth's surface.
Astronauts in orbit do not experience true weightlessness.
True weightlessness never actually occurs. Even in deep space, at vast distances from stars and planets, a very small gravitational field still exists. When there are insignificant gravitational forces acting, objects move as Newton’s first law predicts they should. They either remain at rest or travel in straight lines with constant speed.
The astronauts are in a state of apparent weightlessness.
This occurs when things move with an acceleration, as determined by the gravitational field, i.e. when objects are in free-fall. If you were in an elevator and the cable snapped, you would be in free-fall. You and the elevator would accelerate downwards at 9.8 m s−2. The normal force between you and the floor would be zero. If you held a coin
in front of your nose and released it, it too would accelerate down at 9.8 m s−2and remain in front of your nose! In the frame of reference of the elevator, it would seem as though gravity had been ‘switched off’.
At an altitude of 395 km, where the gravitational field strength is 8.7 N kg−1, the astronauts and their spacecraft orbit at a speed that gives them a centripetal acceleration of 8.7 m s−2. This situation is no different from the earlier situation of falling freely in the elevator. When both you and the elevator were falling at 9.8 m s−2—the rate deter-mined by the gravitational field—you were in a state of apparent weightlessness. The astronauts and their space-craft all have an acceleration of 8.7 m s−2—the rate determined by the gravitational field strength at their altitude—towards the Earth. There is no normal force between the astronauts and the floor (or ceiling!) of their spacecraft. They too are in a state of apparent weightless-ness as they orbit the Earth. The only difference is that the astronauts have a velocity that is directed at 90° to their acceleration.
Weightlessness
PHYSICS IN ACTION
Figure 3.26 (a) This astronaut is in orbit at an altitude of 400 km. The gravitational field strength here is 8.7 N kg−1. She is in free-fall: the only force acting on her is gravity. She moves with an acceleration of 8.7 m s−2towards Earth and is in a continual state of apparent weightlessness. (b) These trainee astronauts are in the ‘vomit comet'. This aeroplane is made to plummet from an altitude of around 12 km. The gravitational field strength here is 9.8 N kg−1. The trainee astronauts are in free-fall; the only force acting on them is gravity. They move with an acceleration of 9.8 m s−2towards Earth and are (for about 20 seconds) in a state of apparent
weightlessness.
(a) (b)
• The total energy of a body moving freely through a gravitational field is constant, although the relative amounts of kinetic energy and gravitational potential energy may change.
• A satellite in a stable circular orbit has a constant amount of both kinetic energy and gravitational potential energy.
• The energy changes of a body moving freely through a gravitational field can be determined from the gravitational force–distance graph for that body.
• The area under a gravitational force–distance graph gives the change in kinetic energy ∆Ekor change in gravitational potential energy ∆Ug of a body, and indicates the work done by the gravitational field.
3.4 QUESTIONS
In these questions, the following data may be used: radius of Earth =6.4 ×106m, mass of Earth =6.0 ×1024kg.
1 Which one of the following statements is correct?
A satellite in a stable circular orbit around the Earth will have:
A varying potential energy as it orbits B varying kinetic energy as it orbits
C constant kinetic energy and constant potential energy.
The following information applies to questions 2–4. The path of a meteor plunging towards the Earth is as follows.
Ignore air resistance when answering these questions.
2 How does the gravitational field strength change as the meteor travels from point A to point D?
3 Discuss any changes in the acceleration of the meteor as it travels along the path shown.
4 Which one or more of the following statements is correct?
A The kinetic energy of the meteor increases as it travels from A to D.
B The gravitational potential energy of the meteor decreases as it travels from A to D.
The total energy of the meteor remains constant.
The following information applies to questions 5 and 6. The graph shows the force on a mass of 1.0 kg as a function of its distance from the centre of the Earth.
5 a Use the graph to determine the gravitational force between the Earth and a 1.0 kg mass 100 km above the Earth’s surface.
b Use the graph to determine the height above the Earth’s surface at which a 1.0 kg mass would experience a gravitational force of 5.0 N.
6 A meteorite of mass 1.0 kg is speeding towards the Earth. When this meteorite is at a distance of 9.5 ×106m from the centre of the planet, its speed is 4.0 km s−1.
a Determine the kinetic energy of the meteorite when it is 9.5 ×106m from the centre of the Earth.
b Calculate the increase in kinetic energy of the meteorite as it moves from a distance of 9.5 ×106m from the centre of the Earth to a point that is 6.5 ×106m from the centre.
c Ignoring air resistance, what is the kinetic energy of the meteorite when it is 6.5 ×106m from the centre of the Earth?
d How fast is the meteorite travelling when it is 6.5 ×106m from the centre of Earth?
A B
C
D
Distance from the centre of the Earth (× 10 6 m)
Force (N)
10 8 6 4 2 0
6.0 7.0 8.0 9.0 10.0
3.4 SUMMARY ENERGY CHANGES IN GRAVITATIONAL FIELDS
In the following questions, you may assume the following: universal constant of gravitation,
G=6.67 ×10−11N m2kg−2;
gravitational field strength on surface of the Earth: g=9.80 N kg−1.
1 The gravitational force of attraction between two alpha particles is equal to
2.94 ×10−59N when they are separated by 1.0 cm. Calculate the mass of an alpha particle.
2 The gravitational force of attraction between Saturn and Dione, a satellite of Saturn, is equal to 2.79 ×1020N. Calculate the orbital radius of Dione. Data:
mass of Dione =1.05 ×1021kg, mass of Saturn =5.69 ×1026kg.
3 A person standing on the surface of the Earth experiences a gravitational force of 900 N.
What gravitational force will this person experience at a height of 2 Earth radii above the Earth’s surface?
4 Two stars of mass M and m are in orbit around each other. As shown in the followng diagram, they are a distance R apart. A spacecraft located at point X experiences zero net gravitational force.
Calculate the value of the ratio M/m.
5 Neptune has a radius of 2.48 ×107m and a mass of 1.02 ×1026kg.
a Calculate the gravitational field strength on the surface of Neptune.
b A 250 kg lump of ice is falling directly towards Neptune.
What is its acceleration as it nears the surface of Neptune?
6 Given that the mass of the Earth is 5.98 ×1024kg and the mean distance from the Earth to the Moon is 3.84 ×108m, calculate the orbital period of the Moon.
Express your answer in days.
7 One of Jupiter’s moons, Leda, has an orbital radius of 1.10 ×1010m.
The mass of Jupiter is equal to 1.90 ×1027kg. Calculate:
a the orbital speed of Leda b the orbital acceleration of Leda c the orbital period of Leda (in
days).
8 a Explain what is meant by a satellite being in a geosynchronous orbit.
b What is the purpose of such an orbit?
c Assuming that the length of a day on Earth is exactly 24 hours, calculate the radius of orbit of a geostationary satellite (mass of Earth = 6.0 ×1024kg).
CHAPTER REVIEW
The following information applies to questions 7–11. A 20-tonne remote sensing satellite is in a circular orbit around the Earth at an altitude of 600 km.
7 Calculate:
a the orbital speed of the satellite b the kinetic energy of the satellite
c the orbital period of the satellite (in seconds).
8 Ground control decides that this orbit is unsuitable and gives the signal for the satellite’s booster rockets to fire.
The satellite now moves to a new circular orbit with an altitude of 2600 km. For this new orbit, calculate:
a the orbital speed of the satellite b the kinetic energy of the satellite
c the orbital period of the satellite (in seconds).
9 Your calculations for questions 7b and 8b should indicate to you that the kinetic energy of the satellite has decreased as it has moved from the lower orbit to the higher orbit.
a What is the value of this decrease in kinetic energy?
b How do you account for this decrease in kinetic energy?
c Is the total energy of the satellite in its higher orbit the same as that in its lower orbit? Explain your answer.
10 Use the following graph to estimate the increase in gravitational potential energy of the satellite as it moved from its lower orbit to its higher orbit.
11 In light of your answer to question 10, what is the amount of energy that the booster rockets delivered to the satellite as it moved to its higher orbit?
12 A communications satellite of mass 240 kg is launched from a space shuttle that is in orbit 600 km above the Earth’s surface. The satellite travels directly away from the Earth and reaches a maximum distance of 8000 km from the centre of the Earth before stopping due to the influence of the Earth’s gravitational field. Use the graph in question 5 to estimate the kinetic energy of this satellite as it was launched.
Distance from centre of the Earth (x 106 m) Force (x 104 N)
9 The planet Mercury has a mass of 3.30 ×1023kg. Its period of rotation about its axis is equal to 5.07 ×106s. For a satellite to be in a synchronous orbit around Mercury, calculate:
a the orbital radius of the satellite
b its orbital speed c its orbital acceleration.
10 The following graph shows the force on a 20 kg rock as a function of its distance from the centre of the planet Mercury. The radius of Mercury is 2.4 ×106m.
A 20 kg rock is speeding towards Mercury. When the rock is 3.0 ×106m from the centre of the planet, its speed is estimated at 1.0 km s−1. Using the graph, estimate:
a the increase in kinetic energy of the rock as it moves to a point that is just 2.5 ×106m from the centre of Mercury b the kinetic energy of the rock
at this closer point
c the speed of the rock at this point.
The following information relates to questions 11–13. The International Space Station orbits Earth at an altitude of 380 km. The GMS-5 satellite has a
much higher orbit at an altitude of 36 000 km. Data: radius of Earth
=6.4 ×106m, mass of Earth
=6.0 ×1024kg.
11 Determine the value of the ratio:
speed of ISS/speed of GMS-5.
12 Determine the value of the ratio:
period of ISS/period of GMS-5.
13 Determine the value of the ratio:
acceleration of ISS/acceleration of GMS-5.
14 Two satellites S1and S2are in circular orbits around the Earth.
Their respective orbital radii are R and 2R. The mass of S1is twice that of S2. Calculate the value of the following ratios:
a orbital period of S1/orbital period of S2
b orbital speed of S1/orbital speed of S2
c acceleration of S1/ acceleration of S2.
15 Earth is in orbit around the Sun.
The Earth has an orbital radius of 1.5 ×1011m and an orbital period of 1 year. Use this information to calculate the mass of the Sun.
The following diagram applies to questions 16–18. It shows the
gravitational force on a wayward satellite as a function of distance from the surface of the Earth.
16 Which of the following units is associated with the area under this graph?
A J B m s−2 C J s D J kg−1
The satellite is no longer in a stable circular orbit and is travelling closer to the Earth.
17 Which one of the following quantities is represented by the shaded area on the graph?
(Ignore air resistance.) A The kinetic energy of the
satellite at an altitude of 600 km.
B The gravitational potential energy of the satellite at an altitude of 600 km.
C The loss in gravitational potential energy of the satellite as it falls to the Earth's surface.
D The increase in gravitational potential energy of the satellite as it falls to the Earth's surface.
18 How much kinetic energy does the satellite gain as it travels from an altitude of 600 km to 200 km altitude?
Height above Earth's surface (× 105 m) Gravitational force (× 103 N)
2 4 6 8 10
EXAM - STYLE QUESTIONS