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18.2 Newton’s law of gravitation
Teacher notes
OCR Physics A
Finding the mass of the Earth, the Sun, and the
black hole at the centre of our galaxy
Specification references

5.2.2c, 5.4.2a, 5.4.3c

M 0.1, 0.2, 2.2, 2.3
Introduction
Students learn how we measure the masses of large objects: the Earth, the Sun, and finally the
super-massive black hole at the centre of our galaxy. They also apply the idea of escape to a
black hole, although the formula applies to escape from any massive object.
Learning outcomes
After completing the worksheet students should be able to:

understand and apply Newton’s law of gravitation

calculate centripetal acceleration.
Teacher notes
In all the questions Newton’s law of gravitation is important. Students should be able to relate the
force from Newton’s law of gravitation to the centripetal acceleration and also to the period of
orbit of a satellite. The equations of relativity are not used, although teachers should be aware
that they exist and that space and time are warped near a black hole. Students may need to be
shown how to calculate escape velocity using the idea of change in gravitational potential
energy, a slight extension from the material in the textbook and specification.
Answers
1
g =
GM
R2
(1 mark)
Mass of the Earth =
gR 2
= 6.0 ´ 1024 kg
G
(1 mark)
(2 marks)
2
( 2p R )
T
2
2
=
GM
R
Mass of the Sun =
(1 mark)
4p 2 R 3
= 1.9 ´ 1030 kg
2
T G
(1 mark)
(2 marks)
3
a
i
17 light-hours  17  60  60  3.00  108
 1.836 
© Oxford University Press 2016
1013
 1.8 
1013 m
(1 mark)
(2 s.f.)
(1 mark)
(2 marks)
http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original.
1
18.2 Newton’s law of gravitation
Teacher notes
OCR Physics A
ii
centripetal acceleration 
v2
R
(1 mark)
(5 ´ 10 )
6
=
2
1.836 ´ 1013
= 1.36 = 1.4ms -2 (2 s.f.)
(1 mark)
(2 marks)
iii
a=
GM
R2
(1 mark)
(
13
aR 2 1.36 ´ 1.836 ´ 10
mass of black hole =
=
G
6.67 ´ 1011
)
2
 6.87  1036  6.9  1036 kg (2 s.f.)
iv
v
b
Mass of black hole 6.87 ´ 1036
=
= 3.6 ´ 106
Mass of the Sun
1.9 ´ 1030
Same order of magnitude as value in article.
The two values do not agree because the orbit is not circular
and because the value of the speed, in particular, is likely to be accurate to only 1
significant figure (although if it is more than 5000 km s−1 then the black hole is even
more massive).
(1 mark)
(1 mark)
(3 marks)
(1 mark)
(1 mark)
(2 marks)
(1 mark)
(1 mark)
(2 marks)
i
GMm
R
(1 mark)
ii
1 2 GMm
mv =
2
R
(1 mark)
2GM
R
(1 mark)
v2 =
v = 3.0 ´108 m s−1
(1 mark)
(Allow follow through from a iii)
iii
This value is the speed of light
and shows that the gas emits radiation when it is outside the event horizon.
(The values of the mass of the star and the size of the radio object are only
estimates. Of course, relativistic equations should only be applied with objects that
are moving close to the speed of light.)
(3 marks)
(1 mark)
(1 mark)
(2 marks)
© Oxford University Press 2016
http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original.
2