NEWTON'S LAW OF UNIVERSAL GRAVITATION

1. The force of gravity

All objects have a tendency to accelerate towards the earth. Thus the earth exerts a force on these objects () - this is called the FORCE OF GRAVITY.

As the earth is a special case of an object, it must be that ANY TWO OBJECTS EXERT AN ATTRACTIVE GRAVITATIONAL FORCE ON ONE ANOTHER. This is an example of "force at a distance", that is, a LONG RANGE FORCE, where the force between a pair of interacting objects is not dependent on any contact between the objects.

The force of gravity acts over huge distances - the earth is kept in orbit around the sun by the mutual interaction of a force of about 3.5x1022 N acting over a distance of some 150 million kilometers.

Newton postulated that this force between two objects was due to the masses of the objects and that it was proportional to the product of the masses of the two objects. He also deduced that gravitational force was inversely proportional to the square of the distance, r, between the centres of the objects. The force is directed along a line joining the centres of the objects

How did Newton deduce that the force of attraction between two bodies was inversely proportional to the square of the distance between them, that is, F ∝ 1/r2?

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2. The universal law of gravitation

Newton formulated the Universal law of gravitation, which states that

Universal Law of Gravitation
"Every particle in the universe attracts every other particle in the universe with a force that is proportional to the product of the masses and inversely proportional to the square of the distance between the particles."

If two masses m1 and m2 are separated by a distance, r, the magnitude of the gravitational force, F, between them is

where G is a proportionality constant which is the same for all objects in the universe. G is called the UNIVERSAL GRAVITATIONAL CONSTANT. Its value was first used in 1873, using data on the earth's density which was determined experimentally by Henry Cavendish in 1797. Its accepted value is G = 6.6726 x 10-11 N·m2·kg-2.

This equation assumes that the masses are "point masses", that is, masses having a negligible physical size. For most practical purposes, the equation will hold true if the distance separating the masses is considerably larger than the objects' physical size, a situation we find applies to astronomical objects.

3. Gravitational fields

A GRAVITATIONAL FIELD exists wherever a mass experiences a force due to the presence, somewhere in space, of another mass. For example, the earth sets up a gravitational field, as shown by the facts that objects are attracted to the earth. Similarly, the sun has a gravitational field that keeps the planets in elliptical orbits around it. The strength of the gravitational field will depend on the distance from the object setting up the field and the mass of that object. In fact, every particle in the universe, however small, sets up a gravitational field, whose strength depends on the distance from the particle and its mass.

We can define the strength of the gravitational field at any point as the force that field exerts on test body of unit mass. Since that force, F is the product of the mass m of the objects and its acceleration in the field, g, setting m = 1 kg will give field strength Fg = g N.kg-1

4. Acceleration due to gravity

The mass of an object which gives rise to its gravitational attraction for other objects, its GRAVITATIONAL MASS, is not necessarily the same quantity as its INERTIAL MASS, which determines the acceleration of an object in response to a force.

If we postulate that the inertial mass and the gravitational mass are the same, then the acceleration due to gravity at the surface of the earth, g, may be calculated by equating the weight of an object to the force exerted on the object by the earth's gravitational field:

where M is the mass and R is the mean radius of the earth, and m is the mass of the object.

Given G = 6.67x10-11 N.m2.kg-2, M = 5.98x1024 kg, and R = 6.38x106 m, we get a calculated value for g = 9.80 m.s2.

From this it is clear that the acceleration due to the earth's gravitational field is independent of the mass of the object. It has the value of 9.78039 m·s-2 at the equator and 9.83217 m·s-2 at the poles (since the earth is somewhat flattened at the poles). A mean value of 9.81 m·s-2 is routinely used, or, for rough calculations, 10 m·s-2 is a convenient value to remember. (See some values for South Africa.)



(Quite tricky!)

5. Weight versus mass

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Additional questions



Values of g

The acceleration due to the earth's gravity, g, is not constant, and varies with the altitude and precise location on the earth's surface. The shape of the earth is that of an ellipsoid, with local "dips" and "bumps". Values of g, may be calculated using complex mathematical techniques based on measurements provided by the Global Positional System (GPS). Some values for South African locations, calculated by the Geological Survey, are shown here on the right.

Location g (m·s-2)
Bloemfontein
Cape Town
Durban
East London
Pretoria
9.7883
9.7962
9.7931
9.7952
9.7858