STAGE 2 PHYSICS
READING
Motion in 2 Dimensions
Key Ideas pg 55-71
Gravitation and Satellites
Key Ideas
Intended Student Learning
Newton’s Law of Universal Gravitation
Any two particles experience mutually attractive
gravitational forces along the line joining them.
Solve problems involving the use of F  Gm1m2 r 2 ,
where F is the magnitude of the gravitational forces,
m1 and m2 are the masses of the particles, r is the
distance between them, and G is the constant of
universal gravitation.
The magnitude of these forces is directly proportional
to the product of the two masses and inversely
proportional to the square of the distance between
them.
Spherically symmetric objects interact gravitationally as
if their masses were located at their centres.
Using proportionality, discuss changes in the
magnitude of the gravitational force on each of the
masses as a result of a change in one or both of the
masses and/or a change in the distance between
them.
Explain that the gravitational forces are consistent with
Newton’s third law.
Using Newton’s law of universal gravitation and second
law of motion, calculate the value of the acceleration
due to gravity g at a planet or moon.
Satellites in Circular Orbits
The gravitational force causes the centripetal
acceleration when a satellite moves in a circular orbit.
For a particular radius of circular orbit there is only one
possible speed for a stable satellite orbit.
Demonstrate an understanding that the speed, and
hence the period, of a satellite moving in a circular orbit
depends only on the radius of the orbit and not on the
mass of the satellite.
Derive the formula v  GM r for the speed v of a
satellite moving in a circular orbit of radius r about a
spherically symmetric mass M , given that its
gravitational effects are the same as if all its mass
were located at its centre.
Solve problems involving the use of the equations
v  GM r and v  2 r T .
Newton’s Theory of Universal Gravitation
Newton was only able to develop the law of universal gravitation by building on the work of previous
scientists. In particular he was indebted to:
Ptolemy (about 85-165) - an Egyptian with a Greek education, who made careful observations of the
planets and determined their distances from the earth by assuming that if a planet moved faster it
must be closer to the earth
Tyco Brahe (1546-1601) - who made extremely accurate observations of the movements of the
moon and planets over many years
Galileo (1564-1642) - whose observations with telescopes showed that the planets moved around
the sun
Johannes Kepler (1571-1630) - who used Tycho Brahe's observations to arrive at a relationship
between the distance of the planets (R) and their periods (T), i.e. the lengths of their years:
𝑅
3
∝ 𝑇
2
Newton used various measurements, which were already known in his time, to determine the
relationship between the force of gravity, the distance between two objects and the mass of the two
objects.
First the measurements suggested that the force of gravity must be inversely proportional to the
square of the distance between two objects, i.e.
𝐹 ∝
1
𝑟2
Newton realized that the size of the force of gravity must also be proportional to the masses of the
two objects, i.e. F ∝ m1 and F ∝ m2
Putting these all together we get:
“There is a mutual force of attraction between any two objects that is directly proportional to the
product of their masses and inversely proportional to the square of the distance between them. The
mutually attractive gravitational forces act along the line joining the centres of mass of the two
objects.”
Gravity and Newton’s Third Law
According to Newton's third law, if the earth is exerting a force of gravity on the moon, then the
moon must exert an equal and opposite force of gravity on the earth as shown below:
This applies to any two objects. The size of the force of gravity is the same (but in opposite
directions) for each object.
Acceleration due to gravity at the earth's surface
Consider the force of gravity acting on an object of mass mo at the earth's surface. Since the earth is
approximately spherical, its mass, M, behaves as if it is concentrated in the centre.
Radius of the earth = 6380 km = 6.38 x 106 m
Mass of the earth = 5.98 x 1024 kg
Sample Question 1
Use Newton’s Universal Law of Gravitation to show that the acceleration due to gravity on the
surface of the Earth is approximately equal to 9.8 ms-2.
Sample Question 2
The force of gravity between two objects is 10 N. What will this force be if
(a) the mass of one object is doubled
(b) the distance between the objects is halved.
Sample Question 3
A cannonball of mass 20 kg is 0.5 m from the cannon of mass 150 kg. What is the force of gravity
between the cannonball and the cannon?
Satellites in Circular Orbits
A satellite is an object which orbits around a planet. The earth has one natural satellite, the moon,
and thousands of artificial satellites.
The orbits of satellites are very close to circular and will be treated as circles in this section.
For a satellite to travel in a circular orbit, a centripetal acceleration must be acting. This is produced
by the force of gravity between the satellite and the planet.
The planets can also be considered to be satellites around the sun, in nearly circular orbits, and their
centripetal acceleration is produced by the sun's gravity.
Since the force of gravity acts from the centre of the masses, it follows that only orbits which share a
common centre with the central body are possible. (Otherwise there would be a component of the
force which ‘pulls’ the satellite back to a such an orbit.
Speed of a Satellite
Since a satellite is assumed to be moving in a circular orbit, the centripetal force is equal to the force
due to gravity.
Sample Question 4
Derive the formula for the speed of a satellite from the centripetal force and universal gravitation
formulae.
Period of a satellite
The period of a satellite can also be shown to be independent of its mass.
This is the same relationship which was determined by Kepler from observations of the movement
of the planets around the sun. So Kepler's relationship is explained by the Law of Universal
Gravitation.
Height of a satellite above a planet's surface
The radius of a satellite's orbit is measured from a
planet's centre, but often knowing the height of the
satellite above the planet's surface is more useful.
The height of a satellite's orbit = the radius of the orbit - the radius of the planet
Sample Question 5
A Satellite orbits the Earth every 90 mins. Find
a) The radius of the satellites orbit
b) The height of the satellites orbit
c) The linear speed of the satellite around the Earth
Earth’s Artificial Satellites
Two types of satellites have proved to be particularly useful.
Geostationary satellites are used in communications and meteorology.
Low-altitude polar satellites are used in meteorology and surveillance of the Earth's surface.
Geostationary Satellites
A geostationary satellite appears to remain stationary over one particular spot on the Earth's
surface.
There are three conditions necessary for this to occur:

The satellite must be orbiting the earth with the same period as the earth's rotation on its
axis, i.e. 23 hours and 56 minutes

The orbit must be from west to east, i.e. in the same direction as the earth's rotation

The orbit must be positioned over the equator - in any other position the point on the
earth's surface under the satellite will vary over time, even if the period of the satellite is
correct.
Sample Question 6
A geostationary satellite can only orbit the earth at one height because its period is fixed at 23 hours
56 minutes. Find the height of a geostationary orbit.
Coverage by Geostationary Satellites
The only areas which cannot be
covered by three geostationary
satellites are those above latitudes
81°N and 81° S.
Since relatively few people live at
these latitudes, the coverage of the
satellites for communications is very
efficient. Communications to the
opposite side of the world can be
achieved by one satellite communicating with another.
The first geostationary satellite was launched in 1964. These satellites are now the basis for
everyday world-wide communications, e.g. the internet, live sport-telecasts, international phone
calls, etc. They are also used to take meteorological pictures for weather broadcasts.
Because of the great height of geostationary satellites, there is a slight delay between a message
being transmitted and received. This can cause some problems which can be overcome by the
second type of satellite to be considered.
Low Altitude Polar Satellites
These satellites have a relatively low height
above the Earth's surface (up to 2000 km) and
periods between approximately 90 minutes and
2 hours.
A satellite with a period of 90 minutes orbits
the Earth 16 times a day and on each orbit
covers a different part of the Earth's surface
because the Earth rotates beneath the
satellite's orbit.
The whole Earth can be viewed and photographed at least once a day. If more frequent viewing is
required, more satellites can be used.
Polar orbits which cover the whole Earth at least once a day are used in weather forecasting and
surveillance of the Earth's surface for such things as changes in vegetation patterns and defense
purposes.
A network of low-altitude satellites is also used in communications, e.g. for mobile phones, because
low-altitude satellites do not 'have the time delay which is a problem with geostationary satellites.
Another important use is to detect bushfires where smoke obscures normal vision and supply the
information to people coordinating firefighting.
Launching Satellites
Satellites have to reach high speeds in their orbits. It is very costly in materials and fuel to launch a
satellite so anything which can be used to assist the launch is important.
The Earth is already rotating from west to east and if a satellite is launched in this direction it has the
same rotational speed as the Earth. This means that less fuel is needed to get the satellite up to its
required orbit. The surface of the Earth is travelling fastest at the equator (about 450 ms-l).
So the ideal place for launching satellites is as close as possible to the equator, with the launch in a
west to east direction. NASA's launch site, Cape Canaveral, is in Florida in the southern USA.
Sites in Northern Australia have also been suggested, but not yet built.
Sample Question 7
A low-altitude polar satellite is orbiting at a height of 500 km above the Earth's surface.
(a) What is the radius of the satellite's orbit?
(b) What is the satellite's linear speed?
(c) What is the period of the satellite (in minutes)?
(d) How many times does the satellite orbit the earth in one 24 hour period?
Modern Satellites and Space Junk
Today, at the beginning of the 21st century, more than 7000 working satellites orbit the earth, and
more are launched at the rate of about one every four days. Space shuttles are also in orbit at
different times. In addition to these, about 160 000 pieces of space junk whirl around in space, the
remains of dead satellites, exploded missiles, chunks of material fallen off spacecraft, etc. This does
not include the remains of meteorites and comets.
Within 20 years it is estimated the number of pieces of junk will grow to at least 300 000.
If one piece of junk hits another piece they break into even more small fragments travelling at up to
100 000 ms -I.
Even tiny fragments can do an enormous amount of damage to the space shuttle and working
satellites. Every year one or two unmanned satellites stop working, probably due to collision with a
piece of space junk.
A piece of junk the size of a 10c coin could completely destroy the space shuttle. Hence it is flown
upside down and angled forwards so that its toughest surface is facing the direction in which it is
travelling and from which junk is most likely to come.