Newton's law of gravitation
The motion of the planets, the moon and the Sun was the interesting subject among the students of Trinity college at Cambridge in England.
Newton was also one among these students. In 1665, the college was closed for an indefinite period due to plague. Newton, who was then 23 years old, went home to Lincolnshire. He continued to think about the motion of planets and the moon. One day Newton sat under an apple tree and had tea with his friends. He saw an apple falling to ground. This incident made him to think about falling bodies. He concluded that the same force of gravitation which attracts the apple to the Earth might also be responsible for attracting the moon and keeping it in its orbit. The centripetal acceleration of the moon in its orbit and the downward acceleration of a body falling on the Earth might have the same origin. Newton calculated the centripetal acceleration by assuming moon's orbit (Fig.) to be circular.
Acceleration due to gravity on the Earth's surface, g = 9.8 m s-2
Centripetal acceleration on the moon, ac = v2/r
where r is the radius of the orbit of the moon (3.84 × 108 m) and v is the speed of the moon.
Time period of revolution of the moon around the Earth,
T = 27.3 days.
The speed of the moon in its orbit, v = 2πr /T
v = = 1.02 × 103 m s−1
Centripetal acceleration, ac = v2/r = 2.7 × 10−3 m s−2
Newton assumed that both the moon and the apple are accelerated towards the centre of the Earth. But their motions differ, because, the moon has a tangential velocity whereas the apple does not have.
Newton found that ac was less than g and hence concluded that force produced due to gravitational attraction of the Earth decreases with increase in distance from the centre of the Earth. He assumed that this acceleration and therefore force was inversely proportional to the square of the distance from the centre of the Earth. He had found that the value of ac was about 1/3600 of the value of g, since the radius of the lunar orbit r is nearly 60 times the radius of the Earth R.
The value of ac was calculated as follows :
Ac/g = (1/r2) / (1/R2) = 1/3600
Ac = 9.8/3600 = 2.7 × 10−3 m s−2
The law states that every particle of matter in the universe attracts every other particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Consider two bodies of masses m1 and m2 with their centres separated by a distance r. The gravitational force between them is
F α m1m2
F α 1/r2
F = G m1m2 / r2
where G is the universal gravitational constant.
If m1 = m2 = 1 kg and r = 1 m, then F = G.
Hence, the Gravitational constant 'G' is numerically equal to the gravitational force of attraction between two bodies of mass 1 kg each separated by a distance of 1 m. The value of G is 6.67 × 10−11 N m2 kg−2 and its dimensional formula is M−1 L3 T−2
Special features of the law
The gravitational force between two bodies is an action and reaction pair.
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