1 1. Which one of the following graphs best represents the variation of the kinetic energy, KE, and of the gravitational potential energy, GPE, of an orbiting satellite with its distance r from the centre of the Earth? 0 0 0 0 0 0 0 0 Energy Energy Energy Energy KE KE KE KE GPE GPE GPE GPE r r r r A. C. B. D. (1) 2. The rings of Saturn are made of rocky particles that orbit the planet. The period T of each particle depends on its distance r from the centre of Saturn. The period T is proportional to r n . Which one of the following is n equal to? A. 1.0 B. 1.5 C. 2.0 D. 3.0 (1)
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1
1. Which one of the following graphs best represents the variation of the kinetic energy, KE, and of the gravitational potential energy, GPE, of an orbiting satellite with its distance r from the centre of the Earth?
0
0
0
0
0
0
0
0
Energy
Energy
Energy
Energy
KE
KE
KE
KE
GPE
GPE
GPE
GPE
r
r
r
r
A.
C.
B.
D.
(1)
2. The rings of Saturn are made of rocky particles that orbit the planet. The period T of each particle depends on its distance r from the centre of Saturn. The period T is proportional to rn. Which one of the following is n equal to?
A. 1.0
B. 1.5
C. 2.0
D. 3.0 (1)
2
3. Which of the following expressions correctly relates the radius R of the circular orbit of a planet round a star to the period T of the orbit?
A. R3 ∝ T2
B. 31R
∝ T2
C. R2 ∝ T3
D. 21
R ∝ T3
(1)
4. The kinetic energy EK of a satellite in orbit varies with its distance r from the centre of a planet of radius R.
Which one of the following graphs best shows the variation of EK with r?
A.EK
00 r=R r
B.EK
00 r=R r
C.EK
00 r=R r
D.EK
00 r=R r
(1)
3
5. Which one of the following correctly relates the radius R of the circular orbit of planets about the Sun to the period T of the orbit?
A. T ∝ R2
B. T ∝ R3
C. T2 ∝ R3
D. T3 ∝ R2 (1)
6. A satellite is in an orbit around the Earth. It is moved to a new orbit that is closer to the surface of the Earth. Which of the following correctly describes the changes in the gravitational potential energy and in the orbital speed of the satellite?
potential energy speed
A. increases increases
B. increases decreases
C. decreases increases
D. decreases decreases
(1)
4
7. Two satellites X and Y are in orbit around the Earth. The orbital radius of satellite X is twice that of satellite Y.
Which of the following correctly gives the ratio
?Y of period orbitalX of period orbital
A. 22
B. 3 4
C. 22
1
D. 3 41
(1)
8. The escape speed from a planet is defined as the speed at which an object must leave the planet’s surface to
A. escape completely from the gravitational field of the planet.
B. enter a geostationary orbit about the planet.
C. escape from the atmosphere of the planet.
D. overcome the gravitational force of the planet. (1)
5
9. A planet is in a circular orbit of radius r about a star. The period of the planet in its orbit is T. A second planet orbits the same star in a circular orbit of radius rS.
Which of the following is a correct expression for the period of the second planet in its orbit about the star?
A. 23
S Trr
B. Trr 2
3
S
C. 32
S Trr
D. 23
S Trr
(1)
10. Two satellites, X and Y, move in circular orbits about the Earth. The orbital period of satellite X is eight times that of satellite Y.
The ratio isY satellite of radius orbitalX satellite of radius orbital
A. 2.
B. 4.
C. 8.
D. 16. (1)
6
11. Planets A and B orbit the same star. The orbital radius of planet B is four times that of planet A.
Which of the following is the magnitude of the ratio
Aplanet for period orbitalBplanet for period orbital
?
A. 4
B. 8
C. 16
D. 64 (1)
12. The escape speed of an object of mass m from a planet of mass M and radius r depends on the gravitational constant and
A. M and r.
B. m and r.
C. M only.
D. M, m, and r. (1)
13. A spacecraft orbits Earth. An astronaut inside the spacecraft feels “weightless” because
A. the gravitational field in the spacecraft is negligible.
B. the Earth exerts equal forces on the spacecraft and the astronaut.
C. the spacecraft and the astronaut have the same acceleration towards the Earth.
D. the spacecraft and the astronaut exert equal and opposite forces on each other. (1)
7
14. Gravitational field strength at a point may be defined as
A. the force on a small mass placed at the point.
B. the force per unit mass on a small mass placed at the point.
C. the work done to move unit mass from infinity to the point.
D. the work done per unit mass to move a small mass from infinity to the point. (1)
15. The Earth is distance RM from the Moon and distance RS from the Sun. The ratio
Sun the todueEarth at thestrength field nalgravitatioMoon the todueEarth at thestrength field nalgravitatio
is proportional to which of the following?
A. 2
2M
SR
R
B. SR
RM
C. 2
2S
MR
R
D. M
S
RR
(1)
8
16. Newton’s law of gravitation for the force F between two point objects of masses M and m, separated by a distance d may be written as
Fd2 ∝ Mm.
The expression may also be used for the force of attraction between the Sun and the Earth, although they are not point masses. This is because
A. the gravitational constant G is not involved in the expression.
B. the force between the Sun and the Earth is very large.
C. the separation of the Sun and the Earth is much greater than their radii.
D. the mass of the Earth is much less than the mass of the Sun. (1)
17. A powered spaceship is moving directly away from a planet as shown below.
planet Spaceship
P
At point P the motors of the spaceship are switched off but the spaceship remains under the influence of the planet. Which one of the following graphs best represents the variation with time t of the velocity v of the spaceship after it passes point P?
v v
v v
t t
t t
0 0
0 0
0 0
0 0
A. B.
C. D.
(1)
9
18. The diagram below shows lines of electric equipotential. The change in potential on moving from one line to the next is always the same. At which point does the electric field strength have its greatest magnitude?
A
B
C
D
(1)
10
19. Which one of the following graphs best shows the variation of the total energy E of a satellite orbiting the Earth with distance r from the centre of the Earth? (The radius of the Earth is R.)
B. E
r0
0
r = R
A. E
r0
0
r = R
D. E
r0
0
r = R
C. E
r0
0
r = R (1)
11
20. Which one of the following statements correctly defines the gravitational potential at a point P in a gravitational field?
A. The work done per unit mass in moving a small mass from point P to infinity.
B. The work done per unit mass in moving a small mass from infinity to point P.
C. The work done in moving a small mass from infinity to point P.
D. The work done in moving a small mass from point P to infinity. (1)
21. The gravitational potential at point X due to the Earth is –7 kJ kg–1. At point Y, the gravitational potential is –3 kJ kg–1.
The change in gravitational potential energy of a mass of 4 kg when it is moved from point X to point Y is
A. 4 kJ.
B. 10 kJ.
C. 16 kJ.
D. 40 kJ. (1)
12
22. An isolated point object has mass M. A second small point object of mass m is placed a distance x from the larger mass.
Which one of the following is a correct expression for the gravitational potential energy of the mass m?
A. x
GM−
B. x
GMm−
C. 2xGM
−
D. 2xGMm
−
(1)
23. A rocket of mass m stands on the surface of planet Mars. Mars has mass M and radius R. The gravitational potential energy of the rocket due to Mars is
A. .RGM−
B. .RGM+
C. .R
GMm−
D. .R
GMm+
(1)
13
24. A point object of mass m is brought from infinity to the point P, a distance r from the centre of an isolated sphere of mass M.
r P
m
M
The work done by the gravitational force in bringing the point object from infinity to point P is
A. .r
MG
B. .r
MmG
C. .r
MG−
D. .r
MmG−
(1)
14
25. The gravitational potential at the surface of Earth is V. The radius of Mercury is about one third the radius of Earth. Earth and Mercury are spheres of the same density and the volume of a sphere is
.34 3rπ
The gravitational potential at the surface of Mercury is
A. .91V
B. .31V
C. 3 V.
D. 9 V. (1)
26. A satellite is in orbit about Earth. The satellite moves to an orbit closer to Earth. Which of the following correctly gives the change in the potential energy and the kinetic energy of the satellite?
change in potential energy change in kinetic energy
A. Decreases Increases
B. Decreases Decreases
C. Increases Increases
D. Increases Decreases
(1)
15
27. The diagram below shows two planets X and Y of masses 2M and M respectively. The centres of the two planets are separated by a distance 2d. Point P is midway between planets X and Y. The mass of each planet may be assumed to be concentrated at its centre.
planet X planet Y
P
2d
The magnitude of the gravitational field strength at point P due to the two planets is
A. zero.
B. .2dGM
C. .22d
GM
D. .32d
GM
(1)
28. The magnitude of the gravitational field strength at the surface of a planet of radius R is 8.0 N kg–1. Which of the following is a correct expression for the gravitational potential at the surface of the planet?
A. R0.8
−
B. 20.8
R−
C. 0.8
R−
D. – 8.0R (1)
16
29. The diagram below shows some lines of equipotential in the region of an electric field.
X Y
Which graph best shows the magnitude E of the electric field strength along the line XY?
E
X Yposition
A. B.
C. D.E
X Yposition
E
X Yposition
E
X Yposition
(1)
17
30. The centres of two isolated spherical stars each of mass M and radius R are separated by a distance d as shown in the diagram below.
X
M
R
M
R
d
The distance d is very large compared to R. Point X is mid-way between the stars. The gravitational potential at point X due to the two stars is
A. .4d
GM−
B. .2R
GM−
C. .d
GM−
D. zero. (1)
31. The gravitational potential at a point P above the surface of a planet is defined as the work done per unit mass in moving a small test mass
A. from point P to the surface of the planet.
B. from the surface of the planet to point P.
C. from point P to infinity.
D. from infinity to point P. (1)
18
32. A satellite is placed in a circular orbit about the Earth.
Which of the following correctly shows the change in the kinetic energy and in the gravitational potential energy of the satellite with increase in the orbital radius?
kinetic energy gravitational potential energy
A. increase decrease
B. increase increase
C. decrease decrease
D. decrease increase (1)
33. A satellite is in orbit about Earth. The satellite moves to an orbit closer to Earth. Which of the following correctly gives the change in the potential energy and the kinetic energy of the satellite?
change in potential energy change in kinetic energy
A. decreases increases
B. decreases decreases
C. increases increases
D. increases decreases
(1)
19
34. Which of the following diagrams best represents the gravitational equipotential surfaces due to two equal spherical masses?
(1)
35. A satellite of mass m and speed v orbits the Earth at a distance r from the centre of the Earth. The gravitational field strength due to the Earth at the satellite is equal to
A. rv .
B. r
v 2
.
C. r
mv .
D. r
mv 2
.
(1)
20
36. Planet X has radius R and mass M. Planet Y has radius 2R and mass 8M.
Which one of the following is the correct value of the ratio
? Yplanet of surfaceat strength field nalgravitatio Xplanet of surfaceat strength field nalgravitatio
A. 4
B. 2
C. 21
D. 41
(1)
37. In Newton’s universal law of gravitation the masses are assumed to be
A. extended masses.
B. masses of planets.
C. point masses.
D. spherical masses. (1)
21
38. The acceleration of free fall of an object of mass m at the surface of Mars is a. The gravitational field strength at the surface of Mars is
A. a.
B. ma.
C. .ma
D. .am
(1)
22
39. Two isolated spheres of masses M and m are held a distance d apart, as shown below.
d
x
mM
Mass M is greater than mass m.
The gravitational field strength g is measured on a line between the two masses. Which graph best shows the variation with distance x from the larger sphere of the magnitude of the field strength g? The Earth’s gravitational field is to be ignored.
00
g
d x00
g
d x
00
g
d x00
g
d x
A. B.
C. D.
(1)
23
40. The mass of Mars is approximately 0.1 times the mass of Earth and its diameter is approximately 0.5 times that of Earth.
What is the approximate gravitational field strength on the surface of Mars?
A. 2N kg–1
B. 4N kg–1
C. 25N kg–1
D. 50N kg–1 (1)
41. This question is about gravitation.
A binary star consists of two stars that each follow circular orbits about a fixed point P as shown below.
star mass P star massM1 M2
R1 R2
The stars have the same orbital period T. Each star may be considered to act as a point mass with its mass concentrated at its centre. The stars, of masses M1 and M2, orbit at distances R1 and R2 respectively from point P.
(a) State the name of the force that provides the centripetal force for the motion of the stars.
(c) The distance between the centre of the Moon and the centre of Earth is about 4.0 × 108 m. The Moon may also be regarded as a point mass situated at its centre. The orbital period of the Moon about the Earth is 2.4 × 106 s.
A space probe is launched from the equator in the direction of the north pole of the Earth. During the launch, the energy E given to the space probe of mass m is
E =e
34RGMm
where G is the Gravitational constant and M and Re are, respectively, the mass and radius of the Earth. Work done in overcoming frictional forces is not to be considered.
(c) Using your answers in (b) and the total energy supplied to the space probe as given in (a), determine the height of the orbit above the Earth’s surface.
(iii) State what will happen to the height of the space probe above the Earth’s surface and to its speed as air resistance gradually reduces the total energy of the space probe.
(c) In total, there are approximately 1029 electrons in the atoms making up a person. Estimate the electrostatic force of repulsion between two people standing 100 m apart as a result of these electrons.
A spacecraft above Earth’s atmosphere is moving away from the Earth. The diagram below shows two positions of the spacecraft. Position A and position B are well above Earth’s atmosphere.
Earth A B
At position A, the rocket engine is switched off and the spacecraft begins coasting freely. At position A, the speed of the spacecraft is 5.37 × 103 m s–1 and at position B, 5.10 × 103 m s–1. The time to travel from position A to position B is 6.00 × 102 s.
(a) (i) Explain why the speed is changing between positions A and B.
(b) The gravitational field strength at the surface of Jupiter is 25 N kg–1 and the radius of Jupiter is 7.1 × 107 m.
(i) Derive an expression for the gravitational field strength at the surface of a planet in terms of its mass M, its radius R and the gravitational constant G.
(b) (i) Derive expressions, in terms of m, G, M and x, for the kinetic energy of the satellite and for the gravitational potential energy of the satellite.
(iv) The frictional forces will change as the orbit of the satellite changes. Suggest and explain the effect on the motion of the satellite of these changing frictional forces.
(b) Neutron stars are very dense stars of small radius. They are formed as part of the evolutionary process of stars that are much more massive than the Sun.
A particular neutron star has radius R of 1.6 × 104 m. The gravitational field strength at its surface is g0. The escape speed ve from the surface of the star is 3.6 × 107 m s–1.
(i) The gravitational potential V at the surface of the star is equal to – g0R. Deduce, explaining your reasoning, that the escape speed from the surface of the star is given by the expression
(c) The period T of rotation of the neutron star is 0.02 s. Use your answer to (b)(ii) to deduce that matter is not lost from the surface of the star as a result of its high speed of rotation.
(a) Kepler’s third law states that the period T of the orbit of a planet about the Sun is related to the average orbital radius R of the planet by the relationship
T2 = KR3
where K is a constant.
(i) Suggest why the law specifies the average orbital radius.
(iii) State an expression for the magnitude F of the force in (ii) in terms of the mass MS, of the Sun, the mass m of the planet, the radius R of the orbit and the universal gravitational constant G.
Saturn has several rings, each of which consists of many small particles that orbit the planet. Saturn may be considered to be a sphere with its mass M concentrated at its centre.
(a) Deduce that, for a particle in one ring moving in a circular orbit of radius R, the linear speed v of the particle in its orbit is given by the expression
(b) One ring, the A ring, has an outer diameter of 2.72 × 108 m. The mass of Saturn is 5.69 × 1026 kg. A particle orbits on the outer edge of this ring. Determine the time for the particle to complete one orbit of Saturn.
(e) A planet has radius R and the acceleration of free fall at its surface is g. The planet may be considered to be a sphere with its mass concentrated at its centre.
Deduce that the escape speed ves is given by the expression
( ).2es gRv =
Explain your working and state one assumption that is made in the derivation.
(b) A satellite of mass m is in a circular orbit around the Earth at height R from the Earth’s surface. The mass of the Earth may be considered to be a point mass concentrated at the Earth’s centre. The Earth has mass M and radius R.
orbit
satellite mass m
R R
Earth mass M
(i) Deduce that the kinetic energy EK of the satellite when in orbit of height R is
(iii) Explain how your answer to (b)(ii) indicates that the satellite will not escape the Earth’s gravitational field and state the minimum amount of energy that must be provided to this satellite so that it does escape.
(b) A meteorite moves towards the Moon from a long distance away.
(i) On the axes below, sketch a graph to show the variation with distance from the centre of the Moon of the gravitational potential of the meteorite as it approaches the Moon. The radius of the Moon is r.
gravitationalpotential
+ve
–ve
r0
distance from centre of Moon
(2)
(ii) The radius r of the Moon is 1.7 × 106 m and its mass is 7.3 × 1022 kg.
Estimate the impact speed with which the meteorite hits the surface of the Moon.
A spherical planet has radius R and mass M. A satellite of mass m orbits the planet with constant linear speed v at a height h above the planet’s surface, as shown below (not to scale).
planet mass Mv
R h
satellite mass m
50
(a) Outline why
(i) although the satellite is moving at constant speed, it is not in equilibrium.
(c) The total energy of the satellite is reduced. Use your expressions in (b) to outline what change, if any, occurs in the radius of the orbit and the speed of the satellite.
(d) The force of friction between the satellite and the atmospheric air increases as the speed of the satellite increases. By reference to your answer in (c), suggest why small satellites will “burn up” as they re-enter the Earth’s atmosphere.