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If an apple drops off a tree it falls to the ground
The Moon is in orbit round the Earth
Spacecraft are attracted towards distant planets
All masses are attracted in the Universe are attracted to all other masses
We have weight when on the Earth

These facts can all be explained by Gravitation. The laws of gravitation were first proposed by Newton around 1700 A.D but in the twentieth century Einstein extended these ideas in this theory of Relativity.

Newton's Law of Gravitation

Kepler's Laws of planetary motion had stated how the planets moved in their orbits but they did not explain why. This was left to Sir Isaac Newton, who was born in Lincolnshire in 1642. Newton was a very gifted man and became Professor of Mathematics at Trinity College Cambridge at the age of 27! His two most famous contributions to Physics were Principia, a book on mechanics, and Optiks, a book about light.

He realised that it was the same type of force that held the Moon in orbit round the Earth as that which made an apple fall from the tree onto the ground. It was also the force of gravitational attraction that that held the planets in orbit around the Sun.

Newton's ideas went as follows:
All objects moving in a circle must have a centripetal acceleration that is directed towards the centre of the circle.
Now we know the radius of the orbit of the Moon round the Earth and the velocity of the Moon. We can work out its acceleration towards the centre of the Earth. This turns out to be 0.0027 m/s2

Newton realized that this acceleration must be due to the pull of gravity of the Earth, but we know that at the Earth's surface the pull of gravity gives all objects an acceleration of 9.81 m/s2

So clearly the effect of gravity gets less the further out from Earth we go.
In fact the radius of the Moon's orbit is 60 times the radius of the Earth and in this distance the gravity has decreased from 9.81 to 0.0027 m/s2. This is 3600 or 60 x 60 times.

Newton used results like the one we have just seen to work out a formula for the attraction between any two objects. He assumed that the pull of gravity depended on the masses of both objects and proposed the following formula for the attraction between two objects of masses, M and m. separated by a distance d.

Force (F) = GMm/d2

The symbol G is the universal constant of gravitation and is a very small number.
In fact G = 6.7 x 10-11 Nm2 kg2 (0.000000000067 Nm2 kg2 )

This law has been called the Universal Law of Gravitation because it has been found to be true for a huge range of forces, from the attraction between the Sun and the Earth to that between two grains of sand.

Example problems
1. What is the force of attraction between two people, one of mass 80 kg and the other 100 kg if they are 0.5m apart?

F = [G x100 x 80]/0.52 = 2.14 X 10-6 N.

This is a very small force but it does increase as the people get closer together! Actually this example is not accurate because Newton's law really only applies to spherical objects, or at least objects so far apart that they can be effectively considered as spherical.

2. What is the force of attraction between the Earth and the Sun?
Mass of the Sun = 2x1030 kg, mass of the Earth = 6x1024 kg, distance from the Earth to the Sun = 1.5x1011 m
F = [Gx2x1030x6x1024]/[1.5x1011]2 = 6.7 x 1011 N     an enormous force!

Newton's Universal Law of Gravitation has been used to explain some important facts in astronomy such as the tides and the motion of comets. The law also explained why it is that the planets do not follow precise orbits but wobble along their path. This is due to the attraction of other planets. It was the irregular motion of Uranus that led to the discovery of the planet Neptune further out in the solar system.

Measurements of the constant of gravitation (G) are very difficult because of the very small forces involved. Methods have been devised using mountains and pendulum bobs.

For a more details of these measurements see:
16-19 Gravitational constant measurement

Two mathematical results can also be deduced from the formula; one is that all bodies that have an inverse square law of attractive force between them must move about each other in elliptical orbits if they are in motion.

We can also use the Universal Law of Gravitation and the formula for the motion in a circle to prove Kepler's Third Law. The law can also be used to find the masses of the planets or the Sun.


© Keith Gibbs 2020