Gravitational potential energy
(1.1) 4 in which 𝐺 =4𝜋 2 𝐾/𝑀 . And this is the force of gravity between the sun and its planet, acting in a straight line between the two. However, there arises an issue about what exactly 'r' is, since planets cannot be treated like particles. Newton proved using calculus that 'r' was the distance between centres of spheres, not the distance between surfaces of spheres.
Verification of formula of gravity
We may wonder if the gravity that causes planets to orbit is the same as the gravity that causes things to fall on earth. The comparison between the acceleration of an object falling near the surface of the earth and the acceleration that makes the moon orbit around the earth can both be measured. The distance from the centre of the moon to the centre of the earth is around 60 times earth's radius. 5 Equation (1.1) shows that the acceleration due to gravity decreases by the square of the distance. In effect, it predicts that the moon's acceleration is 3600 times smaller than g=9.81m/s 2 , which is 0.00273m/s 2 . 6 We can verify this prediction by using again 𝑣 = 2𝜋𝑟/𝑇 , and we have
Using measurable values T=27.3days, r=384000km, and substitute into the equation, we get a=0.00272m/s 2 . This is preferably good with our prediction, so equation (1.1) is verified as applicable to both planets and objects near earth's surface.
In 1687, Issac Newton published Philosophiæ Naturalis Principia Mathematica , in which he generalized gravity to be the attracting force between any two objects, and named this principle as law of universal gravitation
(1.2)
in which 'G' is called gravitational constant. Cavendish was the first person to measure 'G', by indirectly measuring the force between two large, fixed balls of lead and two smaller balls of lead. By measuring how much the fibre gets twisted (by putting a mirror on the fibre and measuring how much the reflected light deviated from its original position), one can measure the strength of the force. 7 Substitute all known values of the experiment, we can obtain the value of 'G', which turns out to be around 6.67×10 −11 𝑁𝑚 2 𝑘𝑔 −2 .
4 Ibid. 5 NSSDCA, n.d. 6 NSSDCA, n.d. 7 Feynman 2011: 108.
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