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18.2 Newton’s law of gravitation
Stretch and challenge
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
In this activity you will learn how we measure the mass of large objects: the Earth,
the Sun, and finally the super-massive black hole at the centre of our galaxy, the
Milky Way. Some of these measurements were made centuries ago in the time of
Newton but the measurements on black holes and confirmation of their existence
are still continuing. You will also apply the idea of escape velocity to a black hole.
Where an object approaches the speed of light then the equations of relativity
should really be used to obtain exact answers. In these questions you will use
Newton’s equations but the answers are surprisingly accurate!
Learning outcomes
After completing the worksheet you should be able to:


understand and apply Newton’s law of gravitation
calculate centripetal acceleration.
Background
Mass of the Earth: All you need to find the mass of the Earth is the acceleration
due to gravity at the Earth’s surface, 9.81 m s−2, the radius of the Earth, 6.4 × 106 m,
and, of course, Newton’s law of gravitation.
Mass of the Sun: We cannot measure the acceleration due to gravity on the
surface of the Sun in the same way as we can on the surface of the Earth. To find
the mass of the Sun we need to know the period of a planet in orbit around the Sun
and its distance from the Sun.
The Earth is 146 million kilometres from the Sun and orbits the Sun in 1 year.
© Oxford University Press 2016
http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
1
18.2 Newton’s law of gravitation
Stretch and challenge
OCR Physics A
Mass of a black hole:
The article below has been adapted from www.space.com.
Surprising observations of a star swiftly orbiting the cloudy heart of the
Milky Way Galaxy have verified with near certainty the existence of a
central black hole, a theoretical object that still eludes direct detection.
Astronomers watched the star for a decade, tracking two-thirds of its path
around the galactic centre. No object has ever before been seen so close
to the centre of any galaxy, nor has any other object previously been
observed making more than a small fraction of its orbital trek around a
galaxy.
"Our work proves that there is indeed a super massive black hole in our
own galaxy," said Rainer Schoedel, a PhD student at the Max-Planck
Institute for Extraterrestrial Physics (MPE) in Germany.
An international team of astronomers photographed the star as it zoomed
around the galactic centre at speeds ultimately exceeding 11 million mph
(5,000 kilometres per second). Early this year, the star flitted precariously
close to the black hole, coming within 17 light-hours, or just three times
the distance from the Sun to Pluto.
The observations rule out nearly all other possible explanations for the
tremendous amount of matter – equal to some 2.6 million Suns – packed
into a tight spot at the centre of our galaxy.
© www.space.com
Questions
You will need to use G = 6.67 ´10-11 Nm2 kg-2 and c = 3.00 ´108 ms-1
1
Find the mass of the Earth.
(2 marks)
2
Find the mass of the Sun.
(2 marks)
3
a
i
Use data from the article to calculate the closest distance from the star to
the black hole.
(2 marks)
© Oxford University Press 2016
http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
2
18.2 Newton’s law of gravitation
Stretch and challenge
OCR Physics A
ii
Calculate the centripetal acceleration of the star moving at 5000 km s−1 at
the closest distance of approach.
(2 marks)
iii Assuming that the star is in a circular orbit around the black hole,
estimate the mass of the black hole.
(3 marks)
iv Hence, show that the mass of the black hole, some 2.6 million Suns, as
stated in the article, is a reasonable estimate.
(2 marks)
v
Apart from using equations that do not take into account relativity, state
one reason why the two values do not completely agree.
(2 marks)
b As gas, dust, and sometimes even stars fall into a black hole, electromagnetic
waves are emitted. Radio waves emitted from the black hole at the centre of the
Milky Way appear to come from an object about 1  1010 m in radius.
It is not possible to escape if an object comes too close to a black hole –
within what is called the event horizon.
i Write down the formula for the work done to remove an object of mass,
m, to infinity if it is a distance R from an object of mass M.
(1 mark)
© Oxford University Press 2016
http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
3
18.2 Newton’s law of gravitation
Stretch and challenge
OCR Physics A
ii
If the object of mass m has enough kinetic energy it can escape the black
hole. Estimate the escape velocity for an object 1 × 1010 m away from the
black hole at the centre of our galaxy. Use the mass of the black hole
calculated in a iii.
(3 marks)
iii Comment on the value you obtained in ii.
(2 marks)
© Oxford University Press 2016
http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
4