1. The document contains multiple choice and structured questions about gravitation. 2. Questions cover topics like gravitational field strength at different heights, gravitational forces between celestial bodies, and equations for orbital motion. 3. Sample problems are worked out relating to gravitational forces, escape velocities, and the distribution of molecular speeds as it relates to atmospheric evolution.

1. Topic 7 Gravitation
1
Topic 7: Gravitation (extra practice)
Multiple Choice Questions
1 The Earth, of radius r, has gravitational field strength g at its surface. What
is the gravitational field strength at a height h above the ground?
A
gr2
r +h( )2
B
gr
r +h( ) C
g r − h( )
r
D
g r −h( )2
r2
2 The gravitational field strength on the surface of a uniform sphere of
diameter d is X. What would be the gravitational field strength on the
surface of a uniform sphere of half the density and of diameter 2d?
A
1
2
X B X C 2X D 4X
3 The Earth experiences gravitational forces from the Sun, mass MS, and
from the Moon, mass Mm. The distances of the Sun and the Moon from the
Earth are rS and rm respectively.
What is the ratio
force on Earth due to the Sun
force on Earth due to the Moon
?
A
Ms
Mm
rs
rm
⎛
⎝⎜
⎞
⎠⎟ B
Ms
Mm
rm
rs
⎛
⎝⎜
⎞
⎠⎟ C
Ms
Mm
rs
rm
⎛
⎝⎜
⎞
⎠⎟
2
D
Ms
Mm
rm
rs
⎛
⎝
⎜
⎞
⎠
⎟
2
4 A certain star of mass M and radius r rotates so rapidly that material at its
equator only just remains on its surface. The period of rotation is
A 2π
r
G
B 2π
G
r
C 2π
r
MG
D 2π
r3
MG
E 2π
MG
r3
5 In two widely-separated planetary systems whose suns have masses S1
and S2, planet P1 of mass M1 and planet P2 of mass M2 are observed to
have circular orbits of equal radii. If P1 completes an orbit in half the time
taken by P2 it may be deduced that
A S1 = 4 S2 and M1 = M2 C S1 = 0.25 S2 and M1 = M2
B S1 = 4 S2 only D S1 = 0.25 S2 only
6 A communication satellite which takes 24 hours to orbit the Earth is
replaced by a new satellite which has twice the mass of the old one. The
new satellite also has an orbit time of 24 hours.
What is the ratio of
radius of orbit of new satellite
radius of orbit of old satellite
?
A
1
2
B
1
1
C
2
1
D
2
1
7 X and Y are two points at R and 2R from the centre of the Earth
respectively, where R is greater than the radius of the Earth. The
gravitational potential at X is -800 kJ kg
-1
. When a mass of 1 kg is taken
from X to Y, the work done on the mass is
A -400 kJ B -200 kJ C +200 kJ D +400 kJ E +800 kJ
Structured Questions
8
Fig. 1
Fig. 1 shows a standard kilogram mass at the surface of the Earth and a
spherical region S of radius 2000 m with its centre 4000 m from the
surface of the Earth. The density of the rock in the region S is 2800 kg m
-3
.
(a) What force does the matter in region S exert on the standard
mass? [3]
(b) If region S consisted of oil of density 900 kg m
-3
instead of rock,
what difference would there be in the force on the standard
mass? [3]
(c) Suggest how gravity meters may be used in oil prospecting?
Find the uncertainty within which the acceleration of free fall
needs to be measured if the meters are to detect the (rather
large) quantity of oil stated in (b). [4]
J91/III/1 (part)

2. Topic 7 Gravitation
2
9 (a) The Earth is considered to be a uniform sphere of radius 6370 km,
spinning on its axis with a period of 24.0 hours. The gravitational
field at the Earth’s surface is identical with that of a point mass of
5.98 x 10
24
kg at the Earth’s centre. For a 1.00 kg mass situated at
the equator,
(i) calculate the gravitational force on the mass,
(ii) determine the force required to maintain the circular path
of the mass,
(iii) deduce the reading on an accurate spring balance
supporting the mass. [6]
(b) Using your answers to (a), state what would be the acceleration
of the mass at the Earth’s surface due to
(i) the gravitational force alone,
(ii) the force as measured on the spring balance. [2]
(c) A student, situated at the Equator, releases a ball from rest in a
vacuum and measures its acceleration towards the Earth’s
surface. He then states that this acceleration is “the
acceleration due to gravity”. Comment on his statement. [2]
J96/III/2 (part)
10 Two stars, of mass M and 4M, are separated by distance d. Each of them
is moving in a circular orbit about their common centre of mass.
(a) Determine the resultant force acting on each of them.
(b) Find their centre of mass.
(c) Determine their angular velocities.
(d) Describe their motion about their common centre of mass.
11 (a) A planet of mass M is orbiting the Sun of mass MS in a circular
path of radius R. Show that the period T of one complete orbit
is
T = 2π
R3
GMS
[3]
(b) Fig. 6 shows two graphs of R
3
against T
2
; one for the moons of
Jupiter and the other for the moons of Saturn. R is the orbital radius
and T is the orbital period.
Fig. 6
Based on the astronomical data shown in Fig. 6 and using the value
1.90 x 10
27
kg as the mass of Jupiter,
(i) determine the experimental value of the universal
gravitation constant G.
(ii) determine the mass of Saturn. [4]

3. Topic 7 Gravitation
3
12 Values for the gravitational potential due to the Earth with a radius of
6.4 x 10
6
m are given below.
Distance from Earth’s
surface / m
Gravitational potential
φ / MJ kg
-1
0
390 000
400 000
410 000
500 000
-62.72
-59.12
-59.03
-58.94
-58.18
(a) Without drawing a graph, use the data to verify that the
gravitational potential φ is inversely proportional to the distance
r from the centre of Earth. [3]
(b) If a satellite of mass 700 kg falls from a height of 400 000 m to
the Earth’s surface, how much potential energy does it lose? [3]
(c) Deduce a value for the Earth’s gravitational field at a height of
400 000 m. [4]
Adapted from J89/II/2
13 (a) Show that a body projected from the Earth, of radius R, with a speed
equal to or greater than the escape speed
v = 2gR
where g is the gravitational field strength at the surface of Earth, will
never return. Give two assumptions necessary for this result to be
valid
(b) The table below gives the approximate values of the radius and
mass of the Earth, Sun and Moon.
Radius / m Mass / kg
Earth
Sun
Moon
6.0 x 10
6
7.0 x 10
8
1.7 x 10
6
6.0 x 10
24
2.0 x 10
30
7.4 x 10
22
(i) Given that the escape speed of Earth is 1.1 x 10
4
m s
-1
,
estimate the escape speeds from the Sun and the Moon.
One theory of atmospheric evolution suggests that the Earth
originally had an atmosphere rich in hydrogen. However as a result
of a major thermal event in which the temperature rose to about
6000 K, the hydrogen concentration then fell to its present very low
level.
(ii) Making reference to the distribution of molecular speeds,
explain how this increase in temperature could have led to a
substantial loss of hydrogen. (The root-mean-square speed of
hydrogen atoms is about 1.2 x 10
4
m s
-1
at 6000 K.)
(iii) The surface temperature of the sun is also about 6000 K but
hydrogen is the most abundant element in the Sun’s
atmosphere. Why do hydrogen atoms escape much less
readily from the Sun than from the Earth?
(iv) On the Moon, the concentrations of all gases are so low that it
has effectively no atmosphere. Suggest an explanation.
N84/I/15
Solutions
1. A 2. B 3. D 4. D 5. B 6. B 7. D
8 (a) 3.91×10−4
N , (b) 2.65 ×10−4
N, (c) Use at different locations to detect
small reduction in the acceleration of free fall, 3 ×10−4
m s−2
9 (a)(i) 9.83 N (ii) 0.0337 N (iii) 9.80 N, (b)(i) 9.83 m s
-2
(ii) 9.80 m s
-2
(c) Measured value is the acceleration of free fall of 9.80 m s
-2
, not the
acceleration due to gravity 9.83 m s
-2
.
10 (a) F = 4GM2
/ d2
, (b) 0.8d , (c) Equal at ω = 5GM / d3
, (d) Both move in
circular orbit about the centre of mass with 4M having a smaller radius of
0.2d. Both have the same angular velocity,
11 (a) Derive Kepler’s 3
rd
Law (b)(i) 6.93 ×10−11
N m2
kg−2
(ii) 5.34 ×1026
kg
12 (a) φ ∝1/ r ⇒φr = constant , check product of potential and distance
(b) 2.58 GJ, (c) g ≈ Δφ / Δr = 9.0 N kg−1
13 (a) Kinetic energy at the surface of Earth must be larger than the change in
gravitational potential of body moving from surface of Earth to infinity
(b) (i) 5.9 ×105
m s−1
(sun), 2.3 ×103
m s−1
(moon)
(ii) Average kinetic energy of hydrogen atoms is higher than the escape
speed. Many possess sufficient kinetic energy to escape leaving the
lower kinetic energy atoms.
(iii) Escape speed on sun is higher than the average speed of atoms.
Most don’t possess sufficient kinetic energy to escape.
(iv) At its formation, Moon temperature could be very high. Since
escape speed on moon is low, most gasses will escape.