After stating the laws of the motion of bodies, Newton wondered what was the force responsible for the elliptical orbits described by the planets around the Sun. In his book “Philosophiæ Naturalis Principia Mathematica“, published in 1687, he formulates the idea that the force that keeps the planets in their orbit must be inversely proportional to the square of the distance that separates them from the Sun, but also that this same force is responsible for the orbit of the Moon around the Earth.
The gravitational force exerted mutually by two bodies of mass m1 and m2 is given by Newton’s law of universal gravitation:
where r is the distance between the two masses (when they are two large bodies far apart from each other, between their centers), and G a constant of proportionality which, in the units of the International System, has the value:
G = 6.673×10−11 N m2 / kg2
G is called the universal gravitational constant and was measured by Cavendish in 1798, 71 years after Newton’s death.
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The constant G is universal because it has the same value everywhere: the gravitational force between two given masses located at the same distance will have the same value regardless of their location.
Gravitation has no known limit to its range because it does not cancel for any finite value of r. It follows from the universal law of gravitation that the greater the distance between masses, the lower the gravitational force between them.
In addition, the gravitational force is always attractive and acts in the radial direction.
The previous figure shows the gravitational force exerted by the Earth on the Moon (F12). Since Newton’s third law holds, the gravitational force exerted by the Moon on Earth (F21) must have the same magnitude and be oriented in the opposite direction.
By using Newton’s second law combined with the gravitational force, it is possible to show that the Earth’s trajectory is an ellipse in its motion around the Sun. This force explains the motion of all the planets around the Sun.
Here you can find Newton’s laws problems.