You’ve probably heard the story of Newton’s apple. One day, Newton was sitting under his mother’s apple tree when an apple fell off and hit the ground. Newton wondered if there might be some force acting on the apple, pulling it towards the ground. To understand the significance of this simple question, we need to put it in context.

For most of human history, it was just accepted that everything falls. What goes up must come down. To think otherwise would be crazy. Asking why an apple fell to the ground would be a complete waste of time. But by asking why, and whether things really needed to fall, Newton set himself on a line of reasoning that would become the basis for all of classical mechanics.

Back to Newton’s insight. What if the earth was pulling on the apple? What if the apple was pulling on the earth? What if the same force could also pull objects much farther away, like the moon? Newton began to hypothesize that a pulling force existed between every object, big or small, close or distant.

### Deriving Newton’s law of universal gravitation

Gravity isn’t a simple concept to understand. It involves the warping of space-time and requires a background in relativity that we aren’t going to get into today. Despite this, Newton was able to come up with a pretty good approximation for how Gravity works. This approximation is still used today and remains a testament to his genius.

You might recall, from the post on Newton’s laws of motion, that Newton’s second law of motion already provides us with a framework for understanding how most forces work. Net force equals mass times acceleration (F=ma). Newton’s new thesis, that there is a pulling force between objects, could be expected to work in much the same way. Given that there is force pulling the apple and the earth together (F), how do we work out its relationship with mass (m) and acceleration (a)?

Consider the apple. Its acceleration can easily be measured at around 10 meters per second squared. But what if we take something farther away like the moon? The moon is 60x farther away from the center of the earth than the apple is, and its acceleration towards the earth seems almost non-existent. This leads us to our first insight for deriving the law of universal gravitation:

The gravitational force between two objects must get smaller the further apart they are.

Newton went on to specify that the gravitational force is proportional to the distance between the two objects squared.

Now that we know that distance is going to impact gravitational force, we might wonder how mass affects the relationship. In this case the mass wont simply be the mass of the apple but rather the mass of both the apple and the earth. We know there is a strong pull between the apple and the earth, and we know there is almost no pull between the apple and any other apple. This leads us to our next insight:

The greater the masses of the two objects pulling on each other, the stronger the gravitational force will be between them.

We now have a relationship where the gravitational force between two objects is proportional to the product of their masses divided by their distance squared. We can start playing around with our findings. For example, imagine we have two bowling balls sitting on our desk. Both balls have a mass of 10kg and a distance between them of 1m. Solving for gravitational force we get (10*10)/1^2 = 100. 100 newtons of gravitational force? Does this sound right?

Our equation is clearly missing something or everyday objects would be flying towards each other. Newton solved this problem by adding a constant to the equation that would be a very small number. He called this constant G. A century later, Henry Cavendish used new technology to measure what G really was. G = 6.67*(10^-11N)m^2/kg^2 or 0.0000000000667. Redoing our bowling ball example, we now get a very tiny gravitational force, which makes sense. Putting all of this together, we get:

Newton’s law of universal gravitation:

F = G * m1m2 / r^2

F is the force between the masses

G is the gravitational constant

m1 is the first massm2 is the second mass

r is the distance between he centers of the masses

You might recall that the acceleration due to gravity is 9.8s/m^2. We arrive at this number by simply plugging numbers into Newton’s law of universal gravitation. Earth’s mass divided by earths radius squared multiplied by the gravitational constant will equal 9.8m/s^2.

### Revisiting Kepler

You might recall that a few decades before Newton, Johannes Kepler formulated three laws for describing the motion of planets. Those laws were:

Kepler’s laws of planetary motion:

- The obit of a planet is an ellipse with the Sun at one of the two foci.
- A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
- The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

Newton, armed with his law of universal gravitation, his laws of motion, and calculus, was able to prove Kepler’s observations mathematically. He also explained what those laws lacked, by showing how planets pulling on each other would affect their orbits.

### Why satellites don’t fall out of orbit

Now that we are armed with Newton’s law of universal gravitation, we can start to use it to explore some interesting questions.

Why does it appear that gravity has a weaker influence on objects in orbit than on objects on earth’s surface? If we look at the equation, something in orbit and something on the surface of earth are relatively the same distance from the earth’s center. Shouldn’t this imply that the force of gravitation on both objects is nearly the same? So why aren’t satellites falling out of the sky? The answer to this was best expressed by Douglass Adams, writer of *The Hitchhikers Guide to the Galaxy*, who said:

There is an art, it says, or rather, a knack to flying. The knack lies in learning how to throw yourself at the ground and miss.

Satellites are falling towards earth, they are just moving so fast that they keep missing it. If there was zero air drag, this could go on forever. So when we think about objects in orbit, we shouldn’t think of them as floating in a gravity free environment. We should think of them as arrows traveling so fast that they never hit the earth. If we use the centripetal acceleration formula(ac = v^2 / r), we can figure out that the arrow needs to be traveling at 7670m/s to stay in orbit.

If we want to go higher we have to increase our speed, and if we want to go lower we have to decrease our speed. This is how orbital navigation works. So there you have it, objects in orbit and objects on earth’s surface fall in much the same way.

### Conclusion

The amazing thing about Newton, was his ability to think outside of the box. He completely ignored common thinking and asked a question that no one else would, what if objects weren’t just falling but rather being pulled towards each other. This led Newton to formulate a concept of gravity that would last long after he died. However, he never attempted to explain what caused gravity and he even said:

That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and trough which their action and force may be conveyed from one another, is to me so great an absurdity that, I believe, no man who has in philosophic matters a competent faculty of thinking could ever fall into it.

It would be centuries before Einstein arrived and thought even further outside of the box.

References:

https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation

https://www.khanacademy.org/science/physics/centripetal-force-and-gravitationhttps://youtu.be/7gf6YpdvtE0?list=PL8dPuuaLjXtN0ge7yDk_UA0ldZJdhwkoV