Colonel Frank Seely School
3.7.2.2 Grav Field Strength
Q1.The diagram shows an isolated binary star system. The two stars have equal masses, M,
and the distance between their centres is r.
The point P is half-way between the two stars.
What is the gravitational field strength at P?
A
zero
B
−
C
−
D
−
(Total 1 mark)
Q2.In the equation X =
, X represents a physical variable in an electric or a gravitational
field, a is a constant, b is either mass or charge and n is a number.
Which line, A to D, in the table provides a consistent representation of
according to the value of n?
The symbols E, g, V and r have their usual meanings.
n
X
a
b
A
1
E
charge
B
1
V
mass
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X, a and b
Colonel Frank Seely School
C
2
g
G
mass
D
2
V
G
charge
(Total 1 mark)
Q3.A planet has a radius half the Earth’s radius and a mass a quarter of the Earth’s mass. What
is the approximate gravitational field strength on the surface of the planet?
A
1.6 N kg–1
B
5.0 N kg–1
C
10 N kg–1
D
20 N kg–1
(Total 1 mark)
Q4.Two stars of mass M and 4M are at a distance d between their centres.
The resultant gravitational field strength is zero along the line between their centres at a
distance y from the centre of the star of mass M.
What is the value of the ratio
?
A
B
C
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Colonel Frank Seely School
D
(Total 1 mark)
Q5.The gravitational field strengths at the surfaces of the Earth and the Moon are 9.8 N kg –1 and
1.7 N kg–1 respectively. If the mass of the Earth is 81 × the mass of the Moon, what is the
ratio of the radius of the Earth to the radius of the Moon?
A
3.7
B
5.8
C
14
D
22
(Total 1 mark)
Q6.Two stars of mass M and 4M are at a distance d between their centres.
The resultant gravitational field strength is zero along the line between their centres at a
distance y from the centre of the star of mass M.
What is the value of the ratio
?
A
B
C
D
(Total 1 mark)
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Colonel Frank Seely School
Q7.A small mass is situated at a point on a line joining two large masses m1 and m2 such that it
experiences no resultant gravitational force. Its distance from the centre of mass of m1 is r1
and its distance from the centre of mass of m2 is r2.
What is the value of the ratio
?
A
B
C
D
(Total 1 mark)
Q8.Which one of the following gives a correct unit for
A
N m−2
B
N kg−1
C
Nm
D
N
?
(Total 1 mark)
Q9.The gravitational field strength at the surface of the Earth is 6 times its value at the surface
of the Moon. The mean density of the Moon is 0.6 times the mean density of the Earth.
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Colonel Frank Seely School
What is the value of the ratio
A
1.8
B
3.6
C
6.0
D
10
?
(Total 1 mark)
Q10.
Which one of the following statements about gravitational fields is incorrect?
A
Moving a mass in the direction of the field lines reduces its potential energy.
B
A stronger field is represented by a greater density of field lines.
C
D
Moving a mass perpendicularly across the field lines does not alter its potential
energy.
At a distance r from a mass the field strength is inversely proportional to r.
(Total 1 mark)
Q11.The gravitational field strength on the surface of a planet orbiting a star is 8.0 N kg–1. If the
planet and star have a similar density but the diameter of the star is 100 times greater
than the planet, what would be the gravitational field strength at the surface of the star?
A
0.0008 N kg–1
B
0.08 N kg–1
C
800 N kg–1
D
8000 N kg–1
(Total 1 mark)
Q12.
A spherical planet of uniform density ρ has radius R.
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Colonel Frank Seely School
Which line, A to D, in the table gives correct expressions for the mass of the planet and
the gravitational field strength at its surface?
mass of planet
gravitational field
strength at surface
A
B
C
D
(Total 1 mark)
Q13.
A planet has a radius half the Earth’s radius and a mass a quarter of the Earth’s
mass. What is the approximate gravitational field strength on the surface of the planet?
A
1.6 N kg–1
B
5.0 N kg–1
C
10 N kg–1
D
20 N kg–1
(Total 1 mark)
Q14.
The diagram shows two point masses each of mass m separated by a distance 2r.
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Colonel Frank Seely School
What is the value of the gravitational field strength at the mid-point, P, between the two
masses?
A
B
C
D
zero
(Total 1 mark)
Q15.
What would the period of rotation of the Earth need to be if objects at the equator
were to appear weightless?
radius of Earth = 6.4 × 106 m
A
4.5 × 10–2 hours
B
1.4 hours
C
24 hours
D
160 hours
(Total 1 mark)
Q16.
The gravitational potential difference between the surface of a planet and a point P,
10 m above the surface, is 8.0 J kg–1. Assuming a uniform field, what is the value of the
gravitational field strength in the region between the planet’s surface and P?
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Colonel Frank Seely School
A
0.80 N kg–1
B
1.25 N kg–1
C
8.0 N kg–1
D
80 N kg–1
(Total 1 mark)
Q17.
The radius of a certain planet is x times the radius of the Earth and its surface
gravitational field strength is y times that of the Earth.
Which one of the following gives the ratio
A
xy
B
x2y
C
xy2
D
x2y2
?
(Total 1 mark)
Q18.
When at the surface of the Earth, a satellite has weight W and gravitational potential
energy –U. It is projected into a circular orbit whose radius is equal to twice the radius of
the Earth. Which line, A to D, in the table shows correctly what happens to the weight of
the satellite and to its gravitational potential energy?
weight
A
B
C
gravitational potential energy
becomes
increases by
becomes
increases by
remains W
increases by U
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Colonel Frank Seely School
D
increases by U
becomes
(Total 1 mark)
Q19.
A projectile moves in a gravitational field. Which one of the following is a correct
statement for the gravitational force acting on the projectile?
A
The force is in the direction of the field.
B
The force is in the opposite direction to that of the field.
C
The force is at right angles to the field.
D
The force is at an angle between 0° and 90° to the field.
(Total 1 mark)
Q20.
The Earth has density ρ and radius R. The gravitational field strength at the surface
is g. What is the gravitational field strength at the surface of a planet of density 2ρ and
radius 2R?
A
g
B
2g
C
4g
D
16 g
(Total 1 mark)
Q21.
The following data refer to two planets.
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Colonel Frank Seely School
radius/km
density/kg m–3
planet P
8 000
6 000
planet Q
16 000
3 000
The gravitational field strength at the surface of P is 13.4 N kg–1. What is the gravitational
field strength at the surface of Q?
A
3.4 N kg–1
B
13.4 N kg–1
C
53.6 N kg–1
D
80.4 N kg–1
(Total 1 mark)
Q22.
The Global Positioning System (GPS) is a system of satellites that transmit radio
signals which can be used to locate the position of a receiver anywhere on Earth.
(a)
A receiver at sea level detects a signal from a satellite in a circular orbit when it is
passing directly overhead as shown in the diagram above.
(i)
The microwave signal is received 68 ms after it was transmitted from the
satellite. Calculate the height of the satellite.
.............................................................................................................
.............................................................................................................
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Colonel Frank Seely School
(ii)
Show that the gravitational field strength of the Earth at the position of the
satellite is 0.56 N kg–1.
mass of the Earth
mean radius of the Earth
=
=
6.0 × 1024 kg
6400 km
.............................................................................................................
.............................................................................................................
.............................................................................................................
.............................................................................................................
.............................................................................................................
.............................................................................................................
.............................................................................................................
(4)
(b)
For the satellite in this orbit, calculate
(i)
its speed,
.............................................................................................................
.............................................................................................................
.............................................................................................................
.............................................................................................................
.............................................................................................................
(ii)
its time period.
.............................................................................................................
.............................................................................................................
.............................................................................................................
.............................................................................................................
.............................................................................................................
(5)
(Total 9 marks)
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Colonel Frank Seely School
Q23.
A planet has a radius half of the Earth’s radius and a mass a quarter of the Earth’s
mass. What is the approximate gravitational field strength on the surface of the planet?
A
1.6 N kg–1
B
5.0 N kg–1
C
10 N kg–1
D
20 N kg–1
(Total 1 mark)
Q24.
At a distance R from a fixed charge, the electric field strength is E and the electric
potential is V. Which line, A to D, gives the electric field strength and electric potential at a
distance 2R from the charge?
electric field strength
electric potential
A
B
C
D
(Total 1 mark)
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Colonel Frank Seely School
Q25.
A small mass is situated at a point on a line joining two large masses ml and m2 such
that it experiences no resultant gravitational force. If its distance from the mass m1 is r1
and
its distance from the mass m2 is r2, what is the value of the ratio
?
A
B
C
D
(Total 1 mark)
Q26.A planet of mass M and radius R rotates so rapidly that loose material at the equator just
remains on the surface. What is the period of rotation of the planet?
G is the universal gravitational constant.
A
2π
B
2π
C
2π
D
2π
(Total 1 mark)
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Colonel Frank Seely School
Q27.
Which one of the following has different units to the other three?
A
gravitational potential
B
gravitational field strength
C
force per unit mass
D
gravitational potential gradient
(Total 1 mark)
Q28.
The mass of the nucleus of an isolated copper atom is 63 u and it carries a charge
of +29 e. The diameter of the atom is 2.3 × 10–10 m.
P is a point at the outer edge of the atom.
(a)
Calculate
(i)
the electric field strength at P due to the nucleus,
.............................................................................................................
.............................................................................................................
.............................................................................................................
(ii)
the gravitational potential at P due to the nucleus.
.............................................................................................................
.............................................................................................................
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Colonel Frank Seely School
.............................................................................................................
(5)
(b)
Draw an arrow on the above diagram to show the direction of the electric field at the
point P.
(1)
(Total 6 marks)
Q29.The gravitational potential difference between the surface of a planet and a point P, 10 m
above the surface, is 8.0 J kgࢤ 1 . Assuming a uniform field, what is the value of the
gravitational field strength in the region between the planet’s surface and P?
A
0.80 N kgࢤ 1
B
1.25 N kgࢤ 1
C
8.0 N kgࢤ 1
D
80 N kgࢤ 1
(Total 1 mark)
Q30.The following data refer to two planets.
radius / km
density / kg m−3
planet P
8000
6000
planet Q
16000
3000
The gravitational field strength at the surface of P is 13.4 N kg−1. What is the gravitational
field strength at the surface of Q?
A
3.4 N kg−1
B
13.4 N kg−1
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Colonel Frank Seely School
C
53.6 N kg−1
D
80.4 N kg−1
(Total 1 mark)
Q31.Both gravitational and electric field strengths can be described by similar equations written
in the form
(a)
Complete the following table by writing down the names of the corresponding
quantities, together with their SI units, for the two types of field.
symbol
a
gravitational field
quantity
SI unit
electrical field
quantity
SI unit
gravitational
field strength
b
m F–1
c
d
(4)
(b)
Two isolated charged objects, A and B, are arranged so that the gravitational force
between them is equal and opposite to the electric force between them.
(i)
The separation of A and B is doubled without changing their charges or
masses. State and explain the effect, if any, that this will have on the resultant
force between them.
...............................................................................................................
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Colonel Frank Seely School
...............................................................................................................
(ii)
At the original separation, the mass of A is doubled, whilst the charge on A
and the mass of B remain as they were initially. What would have to happen to
the charge on B to keep the resultant force zero?
...............................................................................................................
...............................................................................................................
(3)
(Total 7 marks)
Q32.The gravitational field strength at the surface of a planet, X, is 19 N kg –1.
(a)
(i)
Calculate the gravitational potential difference between the surface of X and a
point 10 m above the surface, if the gravitational field can be considered to be
uniform over such a small distance.
...............................................................................................................
...............................................................................................................
(ii)
Calculate the minimum amount of energy required to lift a 9.0 kg rock a vertical
distance of 10 m from the surface of X.
...............................................................................................................
...............................................................................................................
(iii)
State whether the minimum amount of energy you have found in part (ii) would
be different if the 9.0 kg mass were lifted a vertical distance of 10 m from a
point near the top of the highest mountain of planet X. Explain your answer.
...............................................................................................................
...............................................................................................................
...............................................................................................................
(3)
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Colonel Frank Seely School
(b)
Calculate the gravitational field strength at the surface of another planet, Y, that has
the same mass as planet X, but twice the diameter of X.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
(2)
(Total 5 marks)
Q33.(a)
State, in words, Newton’s law of gravitation.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
(2)
(b)
Some of the earliest attempts to determine the gravitational constant, G, were
regarded as experiments to “weigh” the Earth. By considering the gravitational force
acting on a mass at the surface of the Earth, regarded as a sphere of radius R,
show that the mass of the Earth is given by
where g is the value of the gravitational field strength at the Earth’s surface.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
(2)
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Colonel Frank Seely School
(c)
In the following calculation use these data.
radius of the Moon
gravitational field strength at Moon’s surface
mass of the Earth M
gravitational constant G
= 1.74 × 106 m
= 1.62 N kg–1
= 6.00 × 1024 kg
= 6.67 × 10–11 N m2 kg–2
Calculate the mass of the Moon and express its mass as a percentage of the mass
of the Earth.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
(3)
(Total 7 marks)
Q34.The gravitational field associated with a planet is radial, as shown in Figure 1, but near the
surface it is effectively uniform, as shown in Figure 2.
Alongside each figure, sketch a graph to show how the gravitational potential V
associated with the planet varies with distance r (measured outwards from the surface of
the planet) in each of these cases.
Figure 1
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Colonel Frank Seely School
Figure 2
(Total 4 marks)
Q35.
(a)
(i)
Explain what is meant by the gravitational field strength at a point in a
gravitational field.
...............................................................................................................
...............................................................................................................
...............................................................................................................
(ii)
State the SI unit of gravitational field strength.
...............................................................................................................
(2)
(b)
Planet P has mass M and radius R. Planet Q has a radius 3R. The values of the
gravitational field strengths at the surfaces of P and Q are the same.
(i)
Determine the mass of Q in terms of M.
(ii)
The figure below shows how the gravitational field strength above the surface
of planet P varies with distance from its centre. Draw on the diagram the
variation of the gravitational field strength above the surface of Q over the
range shown.
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Colonel Frank Seely School
(6)
(Total 8 marks)
Q36.
(a)
State the factors that affect the gravitational field strength at the surface of a
planet.
........................................................................................................................
........................................................................................................................
(2)
(b)
The diagram below shows the variation, called an anomaly, of gravitational field
strength at the Earth’s surface in a region where there is a large spherical granite
rock buried in the Earth’s crust.
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Colonel Frank Seely School
The density of the granite rock is 3700 kg m–3 and the mean density of the
surrounding material is 2200 kg m–3.
(i)
Show that the difference between the mass of the granite rock and the mass
of an equivalent volume of the surrounding material is 5.0 × 1010 kg.
(4)
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Colonel Frank Seely School
The universal gravitational constant G = 6.7 × 10–11 N m2 kg–2. Calculate the
difference between the gravitational field strength at B and that at point A on
the Earth’s surface that is a long way from the granite rock.
(ii)
(4)
(iii)
Add to the diagram above a graph to show how the variation in gravitational
field strength would change if the granite rock were buried deeper in the
Earth’s crust.
(1)
(Total 11 marks)
Q37.For which of the following relationships is the quantity y related to the quantity x by the
relationship
x
y
A
energy stored in a spring
extension of the spring
B
gravitational field strength
distance from a point mass
C
de Broglie wavelength of an electron
momentum of the electron
D
period of a mass-spring system
spring constant (stiffness) of the spring
(Total 1 mark)
Q38.A lunar landing module and its parent craft are orbiting the Moon at a height above the
surface of 6.0 × 105 m. The mass of the lunar module is 2.7 × 103 kg. The graph below
shows the variation of the gravitational force on the module with height above the surface
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Colonel Frank Seely School
of the Moon.
(a)
(i)
Using data from the graph, find the gravitational force on the lunar module
and hence find its speed when its orbital height is 6.0 × 105 m.
(3)
(ii)
Using data from the graph, find the change in gravitational potential energy of
the lunar module as it descends from its orbit to the surface of the Moon.
(3)
(b)
The descent of the lunar module is controlled by a set of rockets. Describe how you
would use the data which you have already calculated to determine the minimum
fuel load which would enable the lunar module to land on the surface of the Moon
and subsequently to rejoin its parent craft in orbit. State what additional information
you would need to know.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
(3)
(Total 9 marks)
Q39.(a)
(i)
Explain what is meant by gravitational field strength.
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Colonel Frank Seely School
...............................................................................................................
...............................................................................................................
...............................................................................................................
(1)
(ii)
Describe how you would measure the gravitational field strength close to the
surface of the Earth. Draw a diagram of the apparatus that you would use.
(6)
(b)
(i)
The Earth’s gravitational field strength (g) at a distance (r) of 2.0 × 107 m from
its centre is 1.0 N kg–1. Complete the table with the 3 further values of g.
g/N kg
–1
r/10 m
7
1.0
2.0
4.0
6.0
8.0
(2)
(ii)
Below is a grid marked with g and r values on its axes. Draw a graph showing
the variation of g with r for values of r between 2.0 × 107 m and 10.0 × 107 m.
(2)
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Colonel Frank Seely School
(iii)
Estimate the energy required to raise a satellite of mass 800 kg from an orbit
of radius 4.0 × 107 m to one of radius 10.0 × 107 m.
(3)
(Total 14 marks)
Q40.Graph 1 shows the variation of electric field strength E with separation r for two point
charges.
Graph 2 shows the corresponding variation of electric potential V with separation.
Which line in the table correctly relates data for the two graphs?
Magnitude of electric field strength at
separation d
Magnitude of electric potential at
separation d
A
Gradient of graph 2 at separation d
Area under graph 1 from separation d to
∞
B
Area under graph 2 from separation d to
∞
Area under graph 1 from separation d to
∞
C
Gradient of graph 2 at separation d
Gradient of graph 1 at separation d
D
Area under graph 2 from separation d to
∞
Gradient of graph 1 at separation d
(Total 1 mark)
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Colonel Frank Seely School
M1.A
[1]
M2.C
[1]
M3.C
[1]
M4.B
[1]
M5.A
[1]
M6.B
[1]
M7.C
[1]
M8.A
[1]
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Colonel Frank Seely School
M9.B
[1]
M10.
D
[1]
M11.
C
[1]
M12.
B
[1]
M13.
C
[1]
M14.
D
[1]
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Colonel Frank Seely School
M15.
B
[1]
M16.
A
[1]
M17.
B
[1]
M18.
B
[1]
M19.
A
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Colonel Frank Seely School
[1]
M20.
C
[1]
M21.
B
[1]
M22.
h (= ct) (= 3.0 × 108 × 68 × 10–3) = 2.0(4) × 107 m (1)
(a)
(i)
(ii)
g = (–)
(1)
6
r (= 6.4 × 10 + 2.04 × 107) = 2.68 × 107 (m) (1)
(allow C.E. for value of h from (i) for first two marks, but not 3rd)
g=
(1)
(= 0.56 N kg–1)
4
(b)
(i)
g=
(1)
v = [0.56 × (2.68 × 107)]½ (1)
= 3.9 × 103m s–1 (1)
(3.87 × 103 m s–1)
(allow C.E. for value of r from a(ii)
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Colonel Frank Seely School
[or v2 =
= (1)
v=
(1)
= 3.9 × 103 m s–1 (1)]
(ii)
=
(1)
= 4.3(5) × 104s (1)
(12.(1) hours)
(use of v = 3.9 × 103 gives T = 4.3(1) × 104 s = 12.0 hours)
(allow C.E. for value of v from (I)
[alternative for (b):
(i)
(1)
= 3.8(6) × 103 m s–1 (1)]
(allow C.E. for value of r from (a)(ii) and value of T)
(ii)
(1)
= (1.90 × 109 (s2) (1)
T = 4.3(6) × 104 s (1)
5
[9]
M23.
C
[1]
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Colonel Frank Seely School
M24.
C
[1]
M25.
C
[1]
M26.D
[1]
M27.
A
[1]
M28.
(a)
(i)
E (=
)=
(1)
= 3.15 × 1012Vm 1 (or (NC 1) (1)
‑
‑
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Colonel Frank Seely School
(ii)
V(= –
) = (–)
(1)
= (–) 6.07 × 10 26 (1) – sign and J kg 1
‑
‑
5
(b)
arrow pointing to the right (1)
1
[6]
M29.A
[1]
M30.B
[1]
M31.(a)
________
gravitational
constant
mass
distance
(from mass to
point)
N kg–1
electric
field
strength
N C–1
(1)
or V m–1
N m2 kg– _______ _______ (1)
2
_
_
kg
charge
m
distance
(from
charge
to point)
C
(1)
(1)
m
(4)
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Colonel Frank Seely School
(b)
(i)
none (1)
both FE and FG ∝
(ii)
(hence both reduced to
[affected equally] (1)
charge on B must be doubled (1)
(3)
[7]
M32.(a)
(i)
19 = (–)
gives ΔV = 190 (1) J kg–1 (1)
(ii)
W(= mΔV) = 9.0 × 190 = 1710J [or mgh = 9.0 × 19 × 10 = 1710J] (1)
(iii)
on mountain, required energy would be less
because gravitational field strength is less (1)
max 3
(b)
g∝
∴ g′ =
(or F ∝
or correct use of F =
) (1)
= 4.75(Nkg–1) (1)
2
[5]
M33.(a)
attractive force between two particles (or point masses) (1)
proportional to product of masses and inversely proportional to
square of separation [or distance] (1)
2
(b)
(for mass, m, at Earth’s surface) mg =
(1)
rearrangement gives result (1)
2
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Colonel Frank Seely School
(c)
(1)
= 7.35 × 1022 kg (1)
(= 0.0123) ∴ 1.23%
3
[7]
M34.
gradient decreases as r increases (1)
V increases as r increases (1)
only negative values of V shown (1)
constant gradient (1)
V increases as r increases (1)
[max 4]
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Colonel Frank Seely School
M35.
(a)
(i)
force per unit mass/force per kg
B1
(ii)
N kg−1 not ms–2 alone
B1
2
(b)
(i)
GM/R2 seen
C1
GMQ/(3R)2 seen
C1
mass of Q = 9M
A1
(ii)
passes through (3R, g) and falls off in curve
M1
two further points checked e.g., (6R,g/4) (12R, g/16)
M1
overall line quality – single smooth line (both Ms for this)
A1
6
[8]
M36.
(a)
the radius/diameter of the planet
not ‘size’
B1
the mass (or density) of the planet
B1
2
(b)
(i)
volume of the granite = 4/3π r3
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Colonel Frank Seely School
or
radius of the granite = 0.2 km (may be seen in an
incorrect equation)
B1
2003 or 4/3π 0.23 or 3.35 × 107 m3
B1
Mass = density × volume used with any density and
their volume
(Volume may be in formula form)
If they use correct volume then either 1.24 × 1011
or 7.37 × 1010 gets the mark)
B1
(3700-2200) × 3.35 × 107 or 1500 × 3.35 × 107 kg
or (1.24 × 1011 –7.37 × 1010) or 5.025 × 1010
or 5.03 × 1010 seen
Condone rounding off early leading to 4.6 × 1010 kg
B1
4
NB
1)the fourth mark is not for 5.0 × 1010 – all working
must be shown
2)those who do not show conversion of radius from
km to m in the
calculation but otherwise correct will get 3
(ii)
Gravitation field strength g = GM/r2
or
uses distance of 0.4 km for r
C1
Substitution for extra field strength
= 6.7 × 10–11 × 5.0 × 1010/(0.4×103)2
Condone r = 0.4 for this mark
C1
Correct substitution for the extra field strength
with correct
powers of 10
C1
2.1 × 10–5 N kg–1 (condone m s–2)
or
1.9 × 10–5 if 4.6 × 1010 carried forward from (i)
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Colonel Frank Seely School
A1
4
(iii)
Correct general shape always below original curve
B1
1
Alternative scheme for different approach to (ii)
(ii)
Gravitation field strength = GM/r2
or
uses distance of 0.4 km for r
C1
Correct substitution for field strength for granite (or soil)
6.7 × 10–11 × 1.24 × 1011/(0.4×103)2 or 6.7 × 10–11 × 7.37 ×
1010/(0.4×103)2
Condone r = 0.4 for this mark
C1
Correct substitution for field strength for soil (or granite)
C1
2.1 × 10–5 N kg–1 (condone m s–2)
A1
4
[11]
M37.C
[1]
M38.(a)
(i)
F = 2500 or 2600 N
B1
F = mv / r
2
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Colonel Frank Seely School
B1
1500 m s–1 / 1480 m s–1
or any further progress towards a solution such as
attempting to use: F = GMm / r2
or ½ mv2 = mgh
or equations of motion
or r = 6 ××105
or F = mv2 / (r + h)
or evidence of time wasted, for example, repeated attempts
at a solution
no unit penalty or s.f. penalty
B1
(3)
(ii)
using the area under the graph between 0 and 6.0 × 105 m
or use of VG = GM / r or PE = GMm / r
C1
between 20 and 22 squares
or 1 square = 108 J
or uses the trapezium rule or rectangle and triangle
(allow if they fail with powers of ten)
or attempts to find the difference between the 2 values of PE
C1
2.0 × 109 J to 2.2 × 109 J
or attempt to complete the calculation
(may be confounded by lack of r)
A1
(3)
(b)
(work done by motors) relates to change in PE or (PE + KE)
B1
2 × (PE + KE) condone 2 × (PE)
B1
need to know energy value of fuel / fuel density / energy density of fuel /
fuel economy / efficiency of engines
B1
(3)
[9]
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Colonel Frank Seely School
M39.(a)
(i)
force per unit mass (allow equation with defined terms)
B1
(1)
(ii)
diagram of method that will work
(pendulum / light gates / solenoid and mechanical gate / strobe
photography / video)
B1
pair of measurements (eg length of pendulum and (periodic) time /
distance and time of fall – could be shown on diagram)
M1
instruments to measure named quantities (may be on diagram)
A1
correct procedure (eg calculate period for range of lengths, measure the
time of fall for range of heights)
B1
good practice – series of values and averages / use of gradient of graph
B1
appropriate formula and how g calculated
B1
(6)
(b)
(i)
evidence of gr2 being used
C1
values of 0.25, 0.11, 0.06(25)
no s.f. penalty here unless values given as fractions
A1
(2)
(ii)
points correctly plotted on grid (e.c.f.)
B1
smooth curve of high quality at least to 10 × 107 m, no intercept on r axis
B1
(2)
(iii)
attempt to use area under curve
B1
evidence of × 800 kg
B1
(4.3 – 5.3) × 109 J
B1
or
use of equation for potential ΔEG = m(g1r1 – g2r2)
B1
evidence of × 800 kg
B1
(4.7 – 4.9) × 109 J
B1
max 2 if assumed values of G and M used
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Colonel Frank Seely School
allow calculation of GM from graph followed by substitution into ΔEG =
MG(m / r1 – m / r2) for 3 marks
(3)
[14]
M40.A
[1]
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Colonel Frank Seely School
E1.This question distracted only 13% of the students away from the correct answer. When the
question was pre-tested only 68% of the responses were correct. Yet it should not have
been too demanding for students to appreciate that the field strength would be zero at the
mid-point between two identical stars.
E2.This question proved to be somewhat easier, despite the rather abstract phrasing of the
stem. 85% of the students knew that only alternative C gave a consistent expression i.e. g
= GM / r2.
E5.In this question there were two further tests of gravitational field strength which candidates
found demanding, with facilities of 46% and 50% respectively. The question required
candidates to find the ratio RE : RM when gE : gM are in the ratio 9.8:1.7 and ME is 81 × MM.
Forgetting to take the square root of (RE /RM)22 when applying the equation g = GM / R2
was probably responsible for the incorrect response of the 27% of the candidates who
chose distractor C.
E6.In this question there were two further tests of gravitational field strength which candidates
found demanding, with facilities of 46% and 50% respectively. The question required
candidates to find the ratio RE : RM when gE : gM are in the ratio 9.8:1.7 and ME is 81 × MM.
Forgetting to take the square root of (RE / RM)2 when applying the equation g = GM / R2 was
probably responsible for the incorrect response of the 27% of the candidates who chose
distractor C. This question was concerned with the position of the point between masses
of M and 4M at which there would be no resultant field strength; distractor D was the
choice of 26% of the candidates.
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Colonel Frank Seely School
E7.This question, which involved determining the position of the point between two masses at
which there would be no resultant gravitational force, was repeated from an earlier
examination. Two thirds of the responses were correct, the most common incorrect one
being distractor D – the inverse of the required expression.
E8.This question was on gravitational effects. Rearrangement of possible units to obtain the
ratio of the quantities g2 / G was required; almost 70% of the candidates could do this
correctly but 20% chose distractor B (N kg-1 instead of N m-2).
E9.This question was more demanding algebraically and involved use of a density value to
determine the ratio of Earth’s radius to the Moon’s radius. Slightly under half of the
candidates chose the correct value; incorrect responses were fairly evenly spread
between the other distractors and the question discriminated poorly. This suggests that
many were guessing.
E10.
This question tested knowledge and understanding of gravitational fields. 57% of the
students selected the required incorrect statement in this question One in five of them
chose distractor A. This may have been caused by them thinking they were supposed to
choose the correct statement, or it may have been caused by a general misunderstanding
of gravitational potential that was also evident in Section B of this paper.
E11.
This question which tested how g is connected to the diameter for two stars of
similar density, was the most demanding question on the test – its facility was only 39%.
Equating mg with GMm / R2 and then substituting (4/3) π R3ρ. for M ought to have shown
that g is proportional to the product Rρ. Consequently, if ρ is taken to be the same, g ∝ R.
Yet 33% of the students suggested that g would be 100 times smaller (distractor A), and
not 100 times bigger, when the diameter was 100 times larger.
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Colonel Frank Seely School
E12.
The algebra required to relate the density of a planet to its mass and gravitational
field strength in this question did not prove to be an obstacle to most students because
79% of them gave the correct combination from the table.
E13.
This question, on the gravitational field strength at the surface of a planet, made
similar mathematical demands to the previous question but was answered more
successfully. The facility was 72%, an improvement of over 10% on the result when this
question last appeared in an examination. The question was also an effective
discriminator.
E14.
This question was about the value of the gravitational field strength at the mid-point
between two equal masses; surprisingly, only 60% of the candidates knew that this would
be zero.
E15.
This question required familiarity with the idea that a body appears to become
weightless when its centripetal acceleration is just equal to the local value of the
acceleration due to gravity. Hence, if this were to happen at the surface of the Earth, ω2R
would have to equal 9.81 m s–2. The question had a facility of 55%, but one in five
candidates selected distractor A.
E16.
This question was a direct test of the equation connecting field strength and potential
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Colonel Frank Seely School
gradient, g = –ΔV/Δr. The outcome from this question was very similar to when it was last
used; the facility was 72% and there were no particularly strong distractors.
E17.
The correct algebraic rearrangement of g = GM/R2 would deliver a correct answer in
this question, achieved by 62% of the candidates.
E18.
This question revived a question used in an Advanced Supplementary examination
almost ten years ago. The topics, gravitational force and gravitational potential energy for
an Earth satellite, were better known in 2006 than by the previous candidates: the facility
increased from 39% in 1997 to 54% on this occasion. Distractor D was chosen by 22% of
the candidates; increasing the gravitational potential energy of the satellite by U would in
fact remove it to infinity. Distractor A was chosen by the 15% of candidates, thinking that
both force and potential are proportional to 1/r.
E19.
The direction of forces in gravitational, electric and magnetic fields continues to be
an area of misunderstanding, as illustrated by the responses in this question, which had a
facility of 55%. Despite the fact that this question was about gravitational fields, just over a
quarter of the candidates selected distractor C, where the force is supposed to be at right
angles to the field. This confusion with a magnetic field is no more understandable than
that of the 11% who chose distractor B, where the force would be in the opposite direction
to the field. Perhaps this latter group were thinking of electrons in an electric field. Such
incorrect responses suggest that candidates were not always reading the questions with
sufficient care.
E20.
The candidates found this question, with a facility of 41%, to be the most demanding
on the test. The pre-test facility of this question had been rather higher. Candidates
continue to have difficulty with algebraic questions such as this, which require two
separate quantities (here field strength and density) to be combined. Practically half of the
responses were divided almost equally between incorrect distractors A and B.
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Colonel Frank Seely School
E21.
This question had been used previously in a linear A level examination, when 62%
gave the correct response. Several linked ideas were necessary to obtain the required
value: the dependence of g on radius and mass, and the connection between mass,
density and volume. Fewer of the 2004 cohort were able to progress through this,
because only 58% responded correctly this time. Since 21% chose distractor A, and 18%
chose C, it is probable that many resorted to guessing.
E22.
Most candidates scored the mark in part (a) (i) and went to use their answer
correctly in part (ii). A small number of candidates however, failed to add the height
calculated in part (i) to the Earth’s radius or added the radius in km to the height in m.
They were usually able to gain some credit for knowing the correct equation to use.
In part (b) (i), many candidates gave a clear and correct expression, using either the
expressions for centripetal acceleration or the speed in terms of the mass of the Earth.
Weaker candidates confused the symbols for speed and gravitational potential on the data
sheet and attempted to calculate the speed using the expression for gravitational
potential. Most candidates who completed part (i) went on to complete part (ii)
successfully, although some lost the final mark as a result of giving the answer to too
many significant figures. Some candidates in part (ii) successfully related the time period
to the radius of orbit and thus gained full credit. A small minority of candidates gained no
credit as a result of misreading part (b), attempting to provide answers based on a time
period of 24 hours.
E23.
The gravitational field strength at the surface of a planet and its relation with radius
and mass was the subject tested by this question. 61% of the candidates selected the
correct response, a 10% improvement over the pre-test facility. Distractor B, the most
common wrong response, was chosen by just over one in five of the candidates.
E24.
This question, with a facility of 66%, examined the variations of electric field strength
and electric potential with distance in a radial field. Distractor D was hardly ever chosen,
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Colonel Frank Seely School
with wrong answers divided mainly between distractors A and B.
E25.
Gravitation was the subject being tested in this question, on the inverse square law,
which had a facility of 61% but did not discriminate as well as it had when used in a
previous AS level examination.
E27.
Gravitation was the subject being tested in this question. The fact that candidates for
the examinations were not familiar with the units of field and potential became clear in the
scripts for Unit 4 Section B, and so it is hardly surprising that the facility of this question
was no higher than 54%.
E28.
There were many pitfalls en route to successful answers to part (a). Most candidates
obtained little reward in this question because they could not steer clear of them.
Examiners were pleased that so many of the candidates were not put off by the slightly
unfamiliar way in which charge was given in part (a)(i), or by the mass given in u in part
(a)(ii). This, at least, showed that some learning is taking place across the topic
boundaries within Module 4. The really serious problems arose with arithmetic, units and
the need to take care in calculations. Typical errors in part (i) were failing to halve the
diameter and forgetting to square the denominator. The unit of electric field strength was
known by some, yet hardly any of the candidates could give a correct unit for gravitational
potential. Carelessness was apparent in the work of all those who omitted the negative
sign from the final value for gravitational potential.
The subject area tested in part (a) remains totally confusing for so many candidates, who
obviously cannot distinguish between the words gravitational and electric or field and
potential. Perhaps they did not read the wording of the question correctly. This may be
more excusable than the huge number of wrong answers to the electric field direction in
part (c): an arrow pointing inwards at P was common, a tangential arrow at P was fairly
frequent, and a vague arrow drawn some distance from P was not exceptional.
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Colonel Frank Seely School
E31.Although part (a) was relatively novel, most candidates could handle the comparison of
gravitational and electric fields. The gaps in the second line of the table could be filled
directly by use of the Data Booklet, but most of the other entries required a little more
thought. Derived units were sometimes quoted (but not accepted) for the electric field
strength: candidates were expected to know that this is N C–1 or V m–1. In the fourth line,
distance (or radius) squared was a surprisingly common wrong answer.
In pan (b)(i) quite a large number of candidates did not state that the resultant force would
be unchanged, even though they had correctly considered the separate effects of a 1 / r2
relationship on both the gravitational and electric forces. The most frequent wrong
response was that the force (presumably the resultant force) would decrease by a factor
of four. In part (b)(ii) many candidates stated that the charge should be increased, without
indicating that it should be doubled – this was expected for the mark to be awarded.
E32.The question was intended to direct the candidates towards g = −(ΔV / Δx) in part (a), but
some preferred to resort to V = −(GM / r). The amalgamation of this equation with (GM /
r2 ) =19 and r =10 led to an apparently correct answer of 190 J kg−1 . However, the physics
of this approach is wrong, because the distance from the surface of planet X to its centre
is certainly not 10 m. Therefore no credit was given unless g = −(ΔV / Δx) was the basis of
the solution. It was seldom possible to award the separate mark allocated for the correct
unit of gravitational potential difference, because very few candidates knew that it is J kg−1
. The straightforward approach to part (a)(ii) is through the direct application of mΔV, but
most candidates preferred to use mgΔh - which was equally acceptable. Most candidates
recognised that the field strength would be slightly less at the top of the highest mountain
of X and therefore gave an acceptable response in part (a)(iii).
In part (b) the ability to distinguish between 2r2 and (2r)2 was beyond the mathematical
skills of the many candidates who, having realised that g ∝ (1 / r2). gave an incorrect
answer of 9.5 N kg−1.
E33.Missing from most attempted statements in part (a) were the expected references to point
masses and to an attractive force. Many candidates simply tried to put the well-known
formula into words, whilst others referred to the sum of the masses rather than the product
of them.
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Colonel Frank Seely School
The equation g = – GM / r2 is given in the Data booklet and mechanical rearrangement of
it leads directly to the expression in part (b). However, this was not what was required by
the wording of the question, and the many candidates who tried this approach were not
given any marks. The acceptable starting point was to equate the gravitational force with
mg.
Answers to part (c) were frequently completely successful, making an interesting contrast
with the earlier parts of this question. The main problems here were omission of kg after
the mass of the moon, significant figure penalties, and arithmetical slips – typically
forgetting to square the denominator.
E35.
(b)
E36.
(a)
(i)
The clear majority of the candidates stated the definition of gravitation
field strength. There was occasional confusion with that of gravitational
potential energy.
(ii)
Most candidates gained this mark a common alternative was ms"2 - which was
penalised as not being the SI unit of gravitational field strength.
(i)
Candidates failing to gain a mass of 9M often used 3R2 rather than
(3R)2. 1/9M was a relatively common incorrect answer.
(ii)
Few gained the full three marks for this part. Most candidates started their
curve from (3R, g) but then went through (4R, g/4) rather than the correct (6R,
g/4).
(a)
Most candidates gave at least one correct factor although vague answers
such as ‘mass’ was not uncommon. The inadequacy of this response was evident
when other candidates referred to the mass of the object at the surface.
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Colonel Frank Seely School
(b)
(i)
This was often very well done although the setting out of the fairly complex
working frequently left much to be desired. It was often difficult to find the
sequence in which the calculations were made. Errors made included use of
the wrong formula and/or the wrong radius and, in the case of weaker
candidates, an incorrect formula for density.
(ii)
Fewer were successful in this part. The use of an incorrect distance (usually
0.2 km) was common. Few appreciated that the use of the given value in part
(i) in the formula field strength = GM/r2 would give the required answer and
most either calculated the value of the field strength for the granite and left it at
that or made other irrelevant calculations using other distance values.
(iii)
This was completed successfully by almost all candidates.
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Colonel Frank Seely School
Resource currently unavailable.
Page 51