# What Really Happened with the Apple?

** **

There is a popular story that Newton was sitting under an apple tree, an apple fell on his head, and he suddenly thought of the Universal Law of Gravitation. As in all such legends, this is almost certainly not true in its details, but the story contains elements of what actually happened. Probably the more correct version of the story is that Newton, upon observing an apple fall from a tree, began to think along the following lines: The apple is accelerated, since its velocity changes from zero as it is hanging on the tree and moves toward the ground. Thus, by Newton’s 2nd Law there must be a force that acts on the apple to cause this acceleration. Let’s call this force “gravity”, and the associated acceleration the “acceleration due to gravity”. Then imagine the apple tree is twice as high. Again, we expect the apple to be accelerated toward the ground, so this suggests that this force that we call gravity reaches to the top of the tallest apple tree**. **

# Sir Isaac’s Most Excellent Idea

Now came Newton’s truly brilliant insight: if the force of gravity reaches to the top of the highest tree, might it not reach even further; in particular, might it not reach all the way to the orbit of the Moon! Then, the orbit of the Moon about the Earth could be a consequence of the gravitational force, because the acceleration due to gravity could change the velocity of the Moon in just such a way that it followed an orbit around the earth.

# Cannon Thought Experiment

- This can be illustrated with the thought experiment shown in the following figure. Suppose we fire a cannon horizontally from a high mountain;
- the projectile will eventually fall to earth, as indicated by the shortest trajectory in the figure, because of the gravitational force directed toward the center of the Earth and the associated acceleration.

# Remember…

- That acceleration is a change in velocity and that velocity is a vector, so it has both a magnitude and a direction. Thus, acceleration occurs if either or both the magnitude and the direction of the velocity change.

# Going back to the cannon thought experiment

- As we increase the muzzle velocity for our imaginary cannon, the projectile will travel further and further before returning to earth. Finally, Newton reasoned that if the cannon projected the cannon ball with exactly the right velocity, the projectile would travel completely around the Earth, always falling in the gravitational field but never reaching the Earth, which is curving away at the same rate that the projectile falls.

# Orbit, you say?

- That is,
*the cannon ball would have been put into orbit around the Earth.*Newton concluded that the orbit of the Moon was of exactly the same nature: the Moon continuously “fell” in its path around the Earth because of the acceleration due to gravity, thus producing its orbit.

# By such reasoning

- Newton came to the conclusion that any two objects in the Universe exert gravitational attraction on each other, with the force having a universal form:

# Big G

- The constant of proportionality G is known as the
*universal gravitational constant*. - It is termed a “universal constant” because it is thought to be the same at all places and all times, and thus universally characterizes the intrinsic strength of the gravitational force.
- G = 6.67 x 10
^{-11}Nm^{2}/kg^{2}

*SchoolWorkHelper*, 2019, https://schoolworkhelper.net/sir-isaac-newton-the-universal-law-of-gravitation/.

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