Newton’s Law of Universal Gravitation

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 ) - 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 ( ) - who made extremely accurate observations of the movements of the moon and planets over many years Galileo ( ) - whose observations with telescopes showed that the planets moved around the sun Johannes Kepler ( ) - 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: 2

Newton’s theory of Universal Gravitation (II) 3

where G = constant of universal gravitation = 6.67 x N.m 2 kg -2 Every particle in the universe attracts every other particle with a force that is proportional to the product of their mass and inversely proportional to the square of the distance between them. This force acts along the line joining the centres of the two particles. 4

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. 5

Acceleration due to gravity at the earth's surface  Consider the force of gravity acting on an object of mass m o 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 10 6 m  Mass of the earth = 5.98 x kg 6

Acceleration due to gravity at the earth's surface (II) The experimentally measured value of the acceleration due to gravity at the surface is confirmed by the law of universal gravitation. 7

Calculations using the law of gravitation  Using ratios without substituting values for all terms  Example 3.1  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. 8

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Substituting values  Example 3.2  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? 10

3.5 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. 11

3.5.1 Impossible orbits 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. 12

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. The speed of a satellite depends only on the radius of its orbit, not on its mass. For a given radius, a stable satellite has only one possible speed. 13

3.5.3 Period of a satellite 14  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.

3.5.4 Height of a satellite above a planet's surface 15 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

Example

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3.6 Earth's artificial satellites 19  Two types of satellites have proved to be particularly useful.  Geostationary satellites - used in communications and meteorology.  Low-altitude polar satellites - used in meteorology and surveillance of the Earth's surface.

3.6.1 Geostationary satellites 20  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.

Height of a geostationary satellite 21  A geostationary satellite can only orbit the earth at one height because its period is fixed at 23 hours 56 minutes  i.e seconds.

22  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 These satellites are now the basis for everyday world-wide communications, e.g. the internet, live sports-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 23

Low altitude polar satellites 24  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.

3.6.3 Launching satellites 25  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.

Example  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?

Answer 27  (a) Use the earth's radius and the satellite's height to find the satellite's radius  Height = 500 km = 500 x 1000 m = 5 x l0 5 m  Radius = height + earth's radius = 5 x l x l0 6  = 6.88 x l0 6  = 6.9 x l0 6 m (2sf)  i.e. the radius of the satellite's orbit is 6.9 x 10 6 m.

(b) Use the satellite's radius to find its speed 28

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Modern satellites and space junk 30  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 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  If one piece of junk hits another piece they break into even more small fragments travelling at up to 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.