The Gravitational Constant
To determine the value of the gravitation constant G it is necessary to measure the force of attraction between two bodies of known masses and known distance separating them. The experiment must have a very high sensitivity since, as mentioned earlier, the gravitational force is extremely weak, i.e. the value of G is very small.
The first accurate measurement of 'G' by Lord Cavendish in England in 1798. The apparatus he used is called as Cavendish torsion balance.
Two small lead balls, each of mass m, are attached to the opposite ends of a light road, which is suspended, with its axis horizontal, by a fine quartz fibre. Two large lead balls each of mass M, are placed on the opposite ends of another rod which is pivoted at the centre, as shown in the figure.
When the large balls are at positions A and B, the small masses are attracted by gravitation, and as a result the fibre is twisted in the counter-clock wise sense as seen from above.
When the large masses are brought to positions A' and B', the fiber is twisted clockwise. The angle Î¸ through which the fiber is twisted when the large balls are moved from one position to the other is measured by observing the deflection of a beam of light reflected from a mirror attached to the fiber.
The Cavendish torsion balance for measuring G |
In a preliminary experiment, by using known forces the deflection of the beam due to the twist of the fiber is calibrated, so that the force per unit twist is known. Multiplying this by angle Î¸ gives the gravitational force F between masses m and M. knowing the masses m and M and measuring the separation r between them, then the value of G can be determined.
Since Cavendish's measurement significant improvements have been made n the determination of G, particularly by pointing and Boys in the 19^{th} century. The present accepted value of G is
G = 6.67 x 10^{-11} N m^{2 }kg^{-2}
Because of the small value of G, the gravitational force between bodies of ordinary sizes is extremely small. The force of attraction between two bodies, each of mass 1 kg separated by a distance of 1 m is only 6.67 x 10-11 N which is too small to be noticed.
That is why we are not aware of any such attraction. The only attraction we notice is the attraction of the earth, because the earth is very massive.
Expression for g
Consider a body of mass m close to the surface of the earth. Then its distance from the centre of the earth will be r = R, the radius of the earth. The force of gravity experienced by the body is
Where M is the mass of the earth. From Newton's second law of motion, the acceleration produced by this force is
'Weighing' the Earth
We can find the mass of the earth from the law of gravitation and the known value of gravitation constant G. The mass of the earth is given by
Now g=9.8ms^{-2} , R=6.38 x 10^{6}m and G=6.67 x 10^{-11} Nm^{2}kg^{-2} .
Using these values, we get
M = 5.94 Ãƒâ€” 10^{24} kg.