FHSST Physics/Forces/Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation

Why does the Earth stay in orbit around the Sun? Shouldn't it fly off tangentially into outer space?

These questions intrigued Newton and inspired his study of gravitation.

Newton realized that a force must be constantly pulling on the Earth, redirecting its motion and preventing it from being flung off. Newton reasoned that this force, which he termed 'gravity', acted between all bodies with mass and varied inversely to the square of the distance between the two bodies.

${\vec {F}}=G\!\cdot \!{\frac {m_{1}m_{2}}{r^{2}}}$

where $G$ is a universal gravitational constant, $m_{1},m_{2}$ are the 2 masses, and $r$ is the distance between the centers of mass.

Newton also realized that this same force which redirects the path of the Earth around the Sun, was also responsible for an apple falling to the ground. In this case, the two masses, the Earth, and the apple, are attracted each other and this exerts a force which pulls the apple towards the center of the Earth. While we can use the Universal Law of Gravitation formula to solve this problem it is often more convenient to realize two facts:

• The Earths mass, $M_{E}$ is constant
• The distance between the apple (or other object) can usually be approximated with just $R_{E}$ , the radius of the Earth, because this is the dominating term.

Thus we can rewrite the equations such that

${\vec {F}}=m{\vec {g}}$

where

$g=G\!\cdot \!{\frac {M_{E}}{R_{E}^{2}}}$

g is the acceleration on Earth and is

$g=-9.8{\frac {m}{s^{2}}}$