Let's look at the connection between the universal gravitation we've been talking about and just the constant acceleration of gravity near the surface of the Earth. Here's the earth yet again, I'm not drawing the continents because I'm really bad at it. Now, the radius of the Earth. Here we go, R_E, and the radius of the Earth is very big, 6.37 times 10_6 meters. The altitude of a commercial plane, which is probably the highest most of us ever get, speaking in terms of elevation, is nine times 10_4 meters. We can see is your entire life is spent within about one percent of R_E. That's true now. This week, Virgin Galactic took a space plane up, SpaceX has taken off putting people and who knows, maybe eventually a lot of us will get to go more than 1 percent. But for now we're all living right around here. The basic idea is that you never experienced a lot of the big fall off. A lot of them have the force varies with one over r squared, you're in such a small range, you don't get to see all that. Let's put the numbers on that. Let's look at the kinematics for a mass m near the earth. We know that that mass m is going to feel F_g equals G, mass of the Earth and that m over really just the radius of the Earth squared. Because it's always at the radius of the Earth and the variation is essentially nothing less than 1 percent. See you say, "I'm going to set that force equal to ma." This is Newton's second law. We're assuming the only force the mass is feeling is gravity. We get that and you say, "I'll just cancel the mass" and you see everything feels the same acceleration, fine. You say, the acceleration then that I'm feeling must be G, mass of the Earth over the radius of the Earth squared. Those are all constants. Just like we say, near the surface of the Earth, the acceleration of gravity is constant, it's always some number, and if you plug in the gravitational constant, mass of the Earth, raise the Earth, indeed you get 9.8 meters per second square. That's where that number comes from in some sense. How big the Earth is, how far we are from the center, and how massive it is. Now you see that this expression is basically what we call g, and you could calculate it for anything. For the moon, for Mars, any time you have a planet and you know the surface, all you need to know is the mass of the planet and how big it is, and that will tell you the gravitational acceleration. One percent is small, but it's not zero. But you actually can detect the change in G as you move up and down. We can look at it graphically here. I'm going to room over there, and then we plot it here. If we look at the universal F_g as a function of radius. The force of it to infinity if the Earth were a point particle and you got right next to it, but really, it's only meaningful outside the surface of the Earth. If you go inside of a planet, we'll get into that some other time. But it comes down like this and it drops off as one over r square that we plotted there. But the point is, you live here, so if we blow that up, it's going down teeny bit, but it is detectable how much it goes down. What you would do to get that is instead of saying, I live only at R_E, is you'd put a small number again. You would say the gravitational constant is G, ME over, and then you'd say, the radius of the Earth plus my altitude h square. That would get you the change in the gravitational constant as you go from altitude zero and you go to higher and higher altitudes. You simply plug in the universal expression. As long as this is small, you're not going to find a very big deviation from 9.8. But there is some deviation that is easily measured.
