[MUSIC] For several thousand years, and probably much longer, people have stared at the planets and stars and wondered, why do they shine? What keeps them moving? Why don't they fall down? So, this week, gravity and how it runs the solar system, the galaxy, and much of the universe. It's a remarkable story, and the people who led us to it have fascinating stories too. Aristarchus of Samos, Tycho Brahe of Hven, Johannes Kepler of Graz, Galileo Galilei of Florence, Isaac Newton of Cambridge, Albert Einstein of Bern, and others. But we'll put the history aside, and go straight to the physics. Newton's first and second law, F total equals ma. It takes a force to accelerate this object whether down, up, or sideways. But changing the direction of velocity is also an acceleration, and it takes force to keep an object traveling in a circle at constant speed. If I remove that force, it travels in a straight line. [SOUND] Newton compared the acceleration of two objects. One of them near to the surface of the Earth at a distance of about 6,400 kilometers from the center, and the other one was the moon at a distance of about 380,000 kilometers from the Earth, and in nearly circular orbit around it. Well, that sounds like a job for you. What are the ratios of their separations and of their accelerations? Let's recap. The apple accelerates at 9.8 meters per second per second, the moon accelerates at R omega squared. And it takes the moon 27.3 days to travel 360 degrees around the Earth. So we write omega equals 2 pi over period and its centripetal acceleration is just 2.7 millimeters per second per second. So, the ratio of the distances is 60 and the inverse ratio of the accelerations is 3,600. Now, because Newton had analyzed orbits, he knew that the force that caused them was proportional to 1 on r squared. So here was his remarkable insight. The forces that govern the moon and the planets are not something mystical and remote. They're familiar. They're the same force that gives objects weight on Earth. So, that calculation you've done is revolutionary, or at least it was in the 17th century. Before Newton, heavenly objects were thought to obey heavenly laws completely different from the laws of our familiar, dirty Earth. For Newton, one law applied everywhere, and it was action at a distance, no medium required. Let's look at that law. To get the accelerations proportional to 1 on r squared, the force has to be proportional to the mass being accelerated. The moon, in our example. Further, for Newton's third law to hold, there had to be an equal and opposite force on the Earth. So, the mutual force must be proportional to both objects involved. Nicely symmetrical as the third law requires. We now call this law Newton's Law of Universal Gravitation and here's how we write it today. Between bodies 1 and 2, the gravitational force f 1 2 is minus big G times m1m2 on r squared. The constant of proportionality, big G, is called Newton's constant. The minus sign tells us it is attractive. The force is in the minus r direction. And remember that symmetry. And remember that symmetry. My weight, 700 Newtons, is the gravitational force that the earth exerts on me. But I attract the earth with an equal and opposite force. [LAUGH] My gravitational field is weak, but multiply it by the mass of the Earth, and the Earth's weight in my gravitational field is also 700 Newtons up. Symmetry explains why the force is proportional to both masses, but why is it an inverse square law? Well, there is a wonderful, elegant reason, but I'm going to leave you in suspense until you study electrostatics and Gauss' law. So, how big is big G? The answer is, not very. Let's consider a mass m, near the surface of the Earth. It's weight is m times little g. And we've now seen that it's also big G times m times the mass of the Earth over the radius of the Earth squared. Set these expressions for the weight equal, cancel m, and we have big G times mass of the Earth equals little g times the radius squared. Newton knew this product, G, M Earth, but he didn't know either factor. Of course, he could estimate the mass of the Earth. It must have a density of several tons per cubic meter, and he knew its size. But the only way to know it precisely was to measure big G directly. Well, that was first done by Henry Cavendish. Here's how he did it. Two large masses on a rod, suspended on a weak spring. Only a tiny twisting force is required to turn the rod. I put two masses here to twist it one way, then here to twist it the other. Well, you notice that my masses don't seem to be moving. Cavendish used 160 kilogram lead balls and a very weak spring, and still the deflection was only a few millimeters. Newton's constant, big G, turns out to be 6.67 by 10 to the minus 11 Newton meters squared per kilogram per kilogram. [LAUGH] Check those units. G times m1m2 over r squared, yes, Newtons of course. Let's do a quiz to get a feel for the strength of gravity. But one important proviso. In these calculations, I measure the separation between the centers of the objects. This only works exactly if the objects are spherically symmetric, and if you're on the outside. Newton worked that out in his shell theorem. It's not in our syllabus, but we'll give a link to it.
