According to a popular myth, Newton discovered gravity when an apple fell on his head. Newton didn’t discover gravity, but he did postulate that the Moon, like the legendary apple, is being pulled down towards the center of the Earth by the force of gravity. In other words, the Moon is falling. Fortunately, instead of falling down, the Moon is instead falling around the Earth. Newton was the first person to realize that the same force of gravity that causes objects to fall to Earth also holds the Moon and planets in their orbits.

Universal Gravitation

To explain the motions of both falling objects and orbiting planets, Newton postulated a universal force of gravity. Universal means that any two objects with mass (you and the Earth, the Earth and the Moon, the Sun and a planet) are pulled towards each other by the force of gravity. And this force can act at a distance, pulling two objects together across vast distances (the average distance between the Sun and Jupiter, for example, is 778 million kilometers).

Newton used observations to build a mathematical model for gravity. To start, he needed to explain Galileo’s observations about falling objects — that in the absence of air resistance all object fall with the same gravitational acceleration. He knew that more massive objects are more difficult to accelerate than less massive objects.  It is harder to throw a bowling ball than a baseball, and – if they’re moving at the same speed – it’s harder to catch a bowling ball than a baseball.  The moving bowling ball in this example would have more inertia than the baseball.

If the force of gravity were a constant downward force that acted the same on all objects, then a baseball would accelerate downward faster than the more-massive bowling ball, which doesn’t happen.  Of course, Newton also knew that the more massive bowling ball was heavier, it experiences a larger gravitational force. To explain lighter and heavier objects falling side by side (when air resistance is very small), Newton postulated that these two effects – the bowling ball’s greater resistance to acceleration and its greater weight – must cancel each other out.

Gravity also follows Newton’s third law. Two objects pulling on each other gravitationally pull on each other with equal force in opposite directions. This means that more massive objects are not only heavier, they also pull more strongly on other massive objects. This is why an apple would weigh less on the Moon compared to on Earth.

Newton also conjectured that the force of gravity decreases with distance. The Moon itself was evidence for this conjecture. Newton calculated the acceleration of the Moon as it circles Earth and found that it was much lower than the acceleration of bodies falling near the surface of the Earth. Finally, gravity pulls two objects together along a line connecting their centers of mass (for spherically-symmetric bodies like stars and planets, the center of mass is the center of the sphere).

Newton summarized his theory of gravity in an equation

[latex]F_{gravity}=G\frac{m_1m_2}{r^2}[/latex]

In this equation, [latex]m_1[/latex] and [latex]m_2[/latex] are the masses of the two bodies and [latex]r[/latex] is the distance between their centers. For example, for an object on the surface of the Earth, [latex]r[/latex] is the radius of the Earth, or about 6,300 km. The constant [latex]G[/latex] is a universal constant; it is the same for any two objects anywhere in the universe. It’s value is

[latex]G=6.67\times 10^{-11} \frac{N \cdot m^2}{km^2}[/latex]

The gravitational constant is a very small number because gravity is a very weak force. Only when at least one of the two bodies is extremely massive (a planet, star, moon, black hole, etc.) is the force of gravity strong enough to notice. This is why you are stuck to the surface of the Earth, but pens, cups, and other small objects do not stick gravitationally to you.

Notice that this equation gives the size of the gravitational pull between two objects and that this same force pulls (in opposite directions) on both objects. If you find, using this equation, that the size of the gravitational force of the Earth on a ball is 1 Newton, then the gravitational force of the ball on the Earth is also 1 Newton.  (A Newton is the SI unit for force.)

Newton’s universal theory of gravity very nicely explains Galileo’s observations about gravity and falling objects. More massive objects are heavier, they feel the pull of Earth’s gravity more strongly.  They are also more difficult to accelerate, because they have more inertial mass. These two effects cancel each other out exactly, an object that has twice as much mass experiences a gravitational force that is twice as strong resulting in the exact same acceleration due to gravity as a less massive object.

Mathematically, this is captured in by the concept of surface gravity or the acceleration due to gravity near an object’s surface. Surface gravity is given the symbol [latex]g[/latex]. According to Newton’s second law

[latex]g=\frac{F_{gravity}}{m}[/latex]

[latex]g=G\frac{mM}{r^2m}[/latex]

[latex]g=G\frac{M}{r^2}[/latex]

In this second equation, [latex]m[/latex] is used for the mass that is falling and [latex]M[/latex] is the larger mass. For the Earth, [latex]M_{Earth}=5.9722\times10^{24}\rm{kg}[/latex] and [latex]g=9.8\frac{m}{s^2}[/latex]. All objects dropped near the Earth’s surface fall with this acceleration (in the absence of air resistance).

Orbits

To explain the Moon’s motion around the Earth, Newton started with the following thought experiment.  Imagine firing a cannon from the top of a tall mountain, a mountain so tall that it reaches above the atmosphere (no air resistance). When the cannon was fired, the cannonball would fall towards the Earth and eventually land. If the cannon were fired faster, the cannon ball would travel further before landing.  Fired faster still, and the cannonball would pass over the horizon, the Earth’s surface curving out from beneath its fall. Fired at just the right speed, the curve of the falling cannonball would exactly match the curve of the Earth’s surface and the cannon would fall all the way around the Earth, returning to where it started.  This is an orbit.

Let’s think about how Newton’s laws apply to the orbiting Moon.

Without gravity, the Moon would sail off into space.  If gravity were too strong, or the Moon weren’t moving fast enough, the Moon would crash into the Earth.  The Moon orbits because its speed and the force of gravity are well matched.