Gravitation
1985IIB2
(a) Two students have heard about the possibility of placing a communication
satellite in an orbit such that it remains vertically above the same place on the
surface of the Earth. One says the satellite must be so far away that it is not
affected by the Earth's gravity. The other asks whether there is a
communications satellite directly above Hong Kong. What would you say to
each of them to help them understand such satellites ?
(b) Considering the Earth to be a non-rotating sphere. Sir Isaac Newton once argued
that if a cannon-ball were fired horizontally at high enough speed from any
point on the Earth, it would eventually return and strike the cannon from behind.
(i) Given that the speed of the cannon-ball remained constant, what would be
the least time for it to arrive at the cannon again ?
(The radius of the Earth may be taken to be 6.4106 m and the
acceleration due to gravity to be 10 m s--2).
(ii) In reality, there are various reasons why the cannon-ball would not arrive
at the cannon again. Give any three, with brief explanations.
1990-II-7
(a)
(i)
Explain what is meant by 'Parking orbits' for satellites.
(ii) If the radius of the Earth is 6 400 km, and the acceleration due to gravity
at the Earth's surface is 10 ms2, find the height of a satellite which is in a
'parking orbit'. Assume the Earth to be a sphere of uniform density.
1991-II-7
(Given : GME = 4.0  1014 N m2 kg-1 where G = gravitational constant.
ME - mass of earth )
A spacecraft (mass m) has just finished its mission on the moon and is planning to
return to the earth. Figure 7.1, which is not to scale, shows how the gravitational
potential, V, between the surface of the moon and the surface of the earth varies along
the line of centres.
(a)
Explain why there is a point of maximum on the graph.
(b) If the spacecraft is to just succeed in reaching the earth,
(i)
what is the minimum energy per unit mass that has to be
supplied ?
(ii)
What is the minimum speed of the spacecraft at take off ?
(c ) A points A and B, the spacecraft is so far away from the moon that its
gravitational attraction on the spacecraft is negligible. The spacecraft is coasting
through space with its motors off. If it had a velocity of 5 200 m s1 when at
point A, 28 000 km from the centre of the earth, what would be its velocity at
point B, 20 000 km from the centre of the earth ?
(d) As the spacecraft is returning to the earth at a very speed, frictional force
between the spacecraft and the atmosphere would cause its surface to reach a
very high temperature. In order to protect the astronauts, what sort of material
should the spacecraft be made of and why ?
1993-II-7
(a) A rocket is fired to launch a spacecraft with an astronaut into an orbit round the
earth. If the initial acceleration of the rocket is 100 m s2 and the mass of the
astronaut is 60 kg, find the supporting force acting on the astronaut
(i) before the rocket is fired from the earth's surface;
(b)
(ii) during the first few seconds after firing.
Explain briefly why the astronaut should lie down in a bed-shaped seat during
launching.
Now the spacecraft with the astronaut is moving round the earth in a circular
orbit of radius r with its engine turned off.
(i)
(ii)
(c )
Derive an expression for the total mechanical energy E of the spacecraft in
terms of G, ME, m and r, where G = gravitational constant
ME = mass of the earth
m = mass of the
spacecraft
Sketch a graph showing how E varies with r for values of r greater that the
earth's radius RE.
(iii) What is the extra energy required for the spacecraft to escape from the
earth's gravitational field ? Express your answer in term of G, ME, m and
r.
The astronaut inside the orbiting spacecraft is said to be 'weightless'.
A student explains that since the orbit is at a great distance from the earth, the
acceleration due to gravity and the weight of the astronaut are both zero. Do you
agree with the student ? Explain your answer.
1994-I-3(AL)
The value of gravitational potential due to a planet are given in Table 3.1 :
Distance from the surface of the planet / m
0
790 000
800 000
810 000
infinity
(a)
(b)
(c )
(d)
Gravitational potential / J kg1
 62.53  106
 55.66  106
 55.58  106
 55.50  106
0
If the gravitational potential is taken to be zero at an infinite distance from the
planet, the gravitational potential at any point closer to the planet than infinity
will be negative. What property of gravitational force ensures this ?
A satellite of mass 2000 kg is raised from the planet's surface to a height of 800
000 m above its surface.
(i) Find the change in gravitational potential energy of the satellite.
(ii) Calculate the kinetic energy of the satellite when it moves round the
planet in a circular orbit at such a height.
Calculate the minimum speed with which a spacecraft must be fired from the
planet's surface so as to escape from it. (Neglect air resistance and assume the
spacecraft itself is unpowered.)
Using the information from Table 3.1, estimate the gravitational field strength.
(i)
at a height of 800 000 m above the surface of the planet.
(ii)
at the surface of the planet. (Show your working)
1998-IA-1(AL)
A space shuttle crew is sent to service the Hubble Space Telescope H which is in a
circular oibit 6.0105 m above the earth’s surface. The crew succeed in moving the
space shuttle S into the same orbit as H and its thrust rocket are shut down. The
telescope is positioned a few kilometres in front as shown in Figure 1.1. Let G be the
gravitational constant and ME the mass of the earth.
(Given : radius of the earth = 6.4  106 m)
(a)
(b)
(c)
Figure 1.1
What is the apparent weight of an astronaut of mass 75 kg inside the shuttle ?
(1 marks)
(i)
Calculate the value of the gravitational field strength in the
orbit.
(3 marks)
(ii)
Calculate the speed and period of the shuttle in the orbit.
(4 marks)
(i) Show that the total mechanical energy of the shuttle is proportional to
1/r, where r is the radius of its orbit.
(2 marks)
(ii)
In order to overtake the telescope, the shuttle must first spiral into an orbit
of a smaller radius in which it would have a larger angular speed. Using
the result in (c)(i), determine whether the shuttle should increase or
decrease its original speed so that it can get into a lower orbit. Explain
your answer.
(2 marks)
1999-IB-9(AL)
In 1989, the spacecraft ‘Voyager 2’ was sent from the earth to the Neptunian system.
Before this expedition, two satellites of Neptune had been discovered, namely Triton
and Nereid. Six more satellites were discovered after this expedition. Their diameters
d, periods T and orbital radii about Neptune r are as follows:
Satellite
d / km
T/hr
New:
1989 N1
420
26.9
1989 N2
200
13.3
1989 N3
140
8.0
1989 N4
160
9.5
1989 N5
90
7.5
1989 N6
50
7.1
Previously known:
Triton
2720
140.9
Nereid
340
8640
Given : radius of the earth = 6400 km
r/ 103 km
117.6
73.6
52.5
60.0
50.0
48.2
354.3
5500
Calculate the gravitational potential at the earth’s surface.
Find the minimum energy supplied to each kilogram of the mass of the
spacecraft so as to enable it to escape from the earth’s gravitational pull.
Show your working.
(a)
(i)
(ii)
(b)
Suggest a reason why 1989 N1 was not discovered from earth evern though it is
bigger than Nereid.
(c)
It can be shown that the orbital radius r and period T satisfy the relation
r 3 GM

T2 42
(i) What is the major assumption for the validity of the above relation?
(ii) Using this information from the newly discovered satellites 1989 N2 to
1989 N6 of Neptune, plot a suitable graph to find the mass of Neptune.
Show your working.
(Given : Gravitational constant G = 6.7  1011 Nm2 kg2).