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The Law of Gravitation
The basic law of gravitation was developed by Isaac Newton in the 1680s, and it describes the magnitude of the gravitational force that exists between two objects. Let us suppose that we have two point objects with masses m and M, and assume that they are separated by a distance R. The law of gravitation then states that the gravitational force between the two objects is given by:
In words, this states that the magnitude of the gravitational force is proportional to the masses of the objects under consideration, and is inversely proportional to the square of the separation of the objects. Thus, if the distance between the objects is doubled, the gravitational force between them drops by a factor of four.
Note that from Newton's third law, our two masses exert equal but opposite gravitational forces on each other. The following diagram illustrates this idea.
When a constant of proportionality is inserted into the expression above for the gravitational force, we get:
Where G is known as the gravitational constant. The value of G can be obtained by undertaking very delicate and sensitive experiments, and the value is found to be approximately:
G ~ 6.7 10-11 N m2 kg-2.
Thus, if the point objects have a mass of 1 kg and the separation is 1 metre, the gravitational force between them is 6.7 10-11 N, which is an extremely small force, far too small to be measured in the laboratory. Indeed, the gravitational force itself is a very weak force, compared with the other forces that act in the universe, such as the electrostatic force. It is only when we are dealing with objects on an astronomical scale that the gravitational force becomes significant.
The units of the gravitational force are newtons, usually abbreviated to N. The gravitational force on a 1 kg mass at the earth’s surface is approximately 9.8 N. Noting that the radius of the earth is about 6,400 km, the formula above can be used to estimate the mass M of the earth:
That is, the mass of the earth (in kg) is the number 6, followed by 24 zeroes! In fact, this is a very good estimate of the earth’s mass – according to Microsoft Encarta 2007, the mass of the earth to three significant figures is 5.97 1024 kg.
So, we see that the gravitational force can be expressed in terms of a relatively simple formula. However, we do need to remember that the expressions above apply to point masses, and not objects of finite size. However, it can be shown that the gravitational force outside a spherical object is the same as would be experienced if all the mass were concentrated at the centre of the sphere. This is why the calculation above provides a very good estimate for the mass of the earth (which is itself approximately spherical).
The Gravitational Field
In the previous section, we considered the force of gravitation in terms of two masses, each of which exerts a gravitational force on the other. An alternative approach is to think in terms of the gravitational field generated by the presence of a mass in a region of space, and the effect of that field on a second mass in the vicinity of the field.
The notion of the gravitational field is illustrated in the following diagram (adapted from Figure 20.3 in Whelan and Hodgson, Essential principles of Physics, 1985).
In this figure, the presence of the mass M establishes a gravitational field in the vicinity of the mass. When a second mass m is placed within the gravitational field from mass M, it experiences a gravitational attraction towards the mass M. From Newton’s third law, the mass M also feels an equal attraction towards mass m.
In some circles, the gravitational field concept as described above leads to a philosophical difficulty. The difficulty is that the presence of the field can only be detected by using a second object that can feel the gravitational force arising from the gravitational field. The question then arises as to whether the field actually exists in the absence of the “test” object – and because the test object is required to detect the field, there is no way to answer the question. In spite of this, the field concept (both in the treatment of gravitation and electric fields) is a valuable concept because there are various techniques for evaluating gravitational fields from various sizes and shapes of object.
For the gravitational field in the figure above, we can define a quantity called the gravitational field strength g as follows:
The units of the gravitational field strength are N kg-1, or m s-2. The latter unit is the unit of an acceleration, which is why g is often referred to as the “acceleration due to gravity”. The value of g for an object at the earth’s surface is approximately 9.8 N kg-1 or 9.8 m s-2.
Now consider the situation in the following diagram:
In this situation, the mass M is taken to be the earth, and we consider the effect of the gravitational field of the earth on a test mass m at or above the earth’s surface. From the preceding analysis, the gravitational force acting on this mass is mg, where g is the gravitational field strength at the location of the test mass. Now, suppose that no other forces act on the mass m. From Newton’s second law, we can write
Where a is the acceleration of the test mass m towards the earth’s surface. Now, there is a nuance in this equation, as follows. In the expression for gravitational force, the mass that appears is called the “gravitational mass”. In the formulation of Newton’ second law, the mass that appears is called the “inertial mass”. Therefore, the above expression would more correctly be written in the form
All the evidence suggests that the gravitational and inertial masses are identical, and therefore the acceleration of our test mass in the earth’s gravitational field is equal to g, the acceleration due to gravity. That is, if you drop an object from a height, it accelerates back to earth with an acceleration of about 9.8 m s-2. We will return to the question of the equivalence of gravitational and inertial mass in our discussion on the Principle of Equivalence later on.