9.8 Universal gravitation
Imagine that you launch a cannonball straight out of a cannon; it goes a bit, then lands. What happens if you shoot it with a bit more velocity? What if you shoot it really fast? You could shoot it with such a velocity that the ground is dropping out from under it at the same rate that it is falling—it would never land! The cannonball would now be in orbit, as shown below.
Isaac Newton [Public domain], via Wikimedia Commons File:Principia1846-513.png;
In the 17th century, Newton discovered that gravity is the force that keeps objects in orbit, and published it in Principia Mathematica. The implications of this were revolutionary—the same force that keeps things to the ground here governs celestial bodies! Gravity acts between any two bodies with mass; they are attracted towards each other. The magnitude of this force is given by:
FG = G (m1 m2⁄r2)
where m1 and m2 are the masses of the two objects, r is the distance between them, and G is the gravitational constant. Newton did not know what the value of G was; it was determined for the first time in the 1790’s by Henry Cavendish. The value of “Big G” in SI units is G = 6.67×10-11 N · m2 ⁄ kg2.
When you are near the surface of Earth, the distance from you to the center of Earth is the planet’s radius R⊕= 6.371 × 106 m. The mass of Earth is M⊕= 5.972 × 1024 kg. If you have a mass m, this means the force of gravity Earth exerts on you is
FG = G (m M⊕ ⁄ R⊕2) = m (9.81) = mg
This is how we know the value of g, the strength of Earth’s gravity. In fact, you can find out the strength of Earth’s gravity even if you are not close to the surface: use R⊕ + h for any height h above the surface.
Example
flic.kr/p/5P143U