Gravitational force
The force of which we are constantly aware in our daily lives is that which pulls us towards the earth. This is called gravitational force. Sir Isaac Newton came to the conclusion that gravitational force exists between all bodies.
Thus, two stones are not only attracted towards the earth but also attract each other. Normally, we do not notice this force owing to its smallness, although it can be measured with sensetive instruments. Nevertheless, two 50000 t ships lying side by side attract each other with a force of about 18 N.
Newton's law of universal gravitation states that any two particles of matter attract one another with a force which is proportional to the product of their masses and inversely proportional to the sqaure of their distance apart.
F = G m1 m2/r².
where
G is the gravitational constant, approximately equal to 6.674×10-11 N m² kg-2
Strictly, this law applies only when the distance is large compared with the dinentions of the particles.
You may know that Einstein has a theory to explain gravity ( see
Einstein’s principle of gravity
Newton realized that gravitational attraction applied not only to bodies on the earth but was also responsible for holding the moon in its orbit about the earth and also the earth and its fellow planets in their orbits round the sun.
Centripetal force
It is important to grasp the idea that, to keep a body moving in a circle there must be a force on it directed towards the centre. This is called centripetal force. Before Newton's time it was belived that invisible spokes radiated out from the sun and pushed the planets round. Newton's insight into the problem convinced him that a push such as this was not necessary. The planets, carrying their atmospheres with them, go on moving in their orbits because the great vaccum of space offers no opposing force to their motion ( do you remember Newton's first law ?).
Centripetal force is, however, required to produce the continuous change of direction which occurs in the orbit and this is provided by gravitational attraction.
We can try a simple experiment to demostrate centripetal force by securely tying a suitable mass on the end of a string and swinging it round. The pull in the string which is providing the centripetal force can easily be felt and we notice that it varies according to mass, speed, and path radius.
In a laboratory experiment, of course, the circular motion of a mass on a pivoted arm will, if left to itself, rapidly come to rest owing to air resistance and so on. No such resistance is offered to the planets as we have already said; so they continue to move.
Weights as units of force
The weights are used as units of force in elementary work but they are unsuitable for accurate scientific work since the weight of a body depends on where it happens to be relative to the earth.
The earth is not a perfect sphere, but bulges at the equator so that if a body is taken from a pole to the equator its distance from the centre of the earth will increase. Consequently, in accordance with Newton's law, the gravitational pull on it will get less.
However, there is another factor causing the weight to decrease, as we shall now explain.
Relation between total gravitational force and weight
If we stand on a spring weighing machine, the force or weight we exert on it comes from the earth's gravitational attraction on us. But the weighing machine does not measure the total gravitational force.
Owing to the rotation of the earth on its axis we happen to be moving in a circle dependent on our geographical latitude.
Consequently, part of the gravitational attraction has to provide centripetal force required to keep us moving in that circle. The remaining part of the gravitational force simply presses us down on the earth's surface.
This part of the gravitational attraction we call our weight, and this is what the spring weighing machine measures.
Only at the poles, where there is no motion in a circle, would it be true to say that weight is equal to the total gravitational force. We must remember, of course, that the two factors we have mentioned, namely, the bulge effect and the centripetal force are both very small.
The reduction in weight of a body as between pole and equator does not amount to more than about 1/2 per cent of the total gravitational force. We may, therefore, define weight as follows:
The weight of a body with respect to the earth is that force which the body exerts on anything which freely supports it.
To avoid confusion we shall now summarize the terms which will be used in any further discussion ( in classical mechanics, of course) relating to bodies at rest on the earth's surface.
1- By the term force of gravity we mean that part of the total gravitational force which acts on a body and so enables it, in turn, to exert an equal force on its support. This force on the support is called the body's weight.
2- The centripetal force is that part of the total gravitational force which is required to constrain the body to move in its circle of latitude.
3- The sum of the force of gravity and the centripetal force is equal to the total gravitational force as given by Newton's law of gravitation.
Weightlessness
We shall now let our imagination run on a "thought experiment"
Suppose we were to stand on a spring weighing machine and that, by some means, the earth's rotation could be speeded up, carrying its atmosphere with it.
With increasing speed, more and more of the earth's gravitational force would be used in providing the extra centripetal force required in the circumstances. Our weight, which is equal to the difference between the total gravitational force and the centripetal force, would therefore be less.
Consequently the weighing machine would indicate a smaller weight.
If the earth's rotational speed continued to increase, then at a certain critical speed, the necessary centripetal force would be just equal to the total gravitational force. No resultant force would be left over to provide weight and so the weighing machine would read zero. In other words we have become weightless although the full gravitational force still continues to act.
Weightlessness in space vehicles
It is a well-known fact that astronauts experience weightlessness when their spacecraft are in orbit about the earth. In such circumstances, the earth's gravitational pull is just sufficient to provide the centripetal force required for their particular mass, speed, and orbital radius. Mechanically speaking they are in a similar position to the person we have just described on a fast rotating earth.
When an astronaut goes outside his cabin, the situation as far as the earth's pull goes is still the same. There is, of course, gravitational attraction between the astronaut and his cabin in accordance with Newton's law but, owing to the comparatively very small masses of both, the attraction between them is exeedingly small. It is far too small, in fact, to bring him back if he jumped off and so he has to use a life line .