Professor Dave here, I want to tell you about Newton's law of universal gravitation. We learned about Newton's laws of motion, but there's one more law of his to discuss, and it's a big one. It's Newton's law of universal gravitation. In possibly one of the greatest strokes of genius in the history of mankind, Newton looked at the motion of the planets in their nearly circular orbits around the Sun and understood that the centripetal force causing this motion was precisely the same force that causes objects to fall down towards Earth, which we call the gravitational force. In this way he proposed that the planets are in a kind of free fall towards the Sun just the same way that the Apple that mythically hit him on the head to produce this insight was in free fall towards the earth. The objects differ greatly in size but the concept is the same. Newton corroborated this notion with a thought experiment. When a cannon fires a cannon ball, the ball eventually hits the ground. If another cannonball is fired with greater force, it will go a little further before hitting the ground. If a cannonball could be fired with an incredibly immense force, it could produce a speed so great that the ball would never hit the ground, since it would fall at the same rate that Earth's curvature is produced. It would thus always be falling towards the earth but never hitting it. Such an object would be said to be in orbit around the Earth. Of course no cannon can do this, but we have finally achieved this feat with all of our satellites and space stations, which are brought up to orbit on rockets .These are very far from Earth's surface and they are moving with such great speed that they, along with anyone on board, are always falling towards the earth but never hitting it, in a free fall just like Newton's apple. This means they orbit around the Earth indefinitely at a fixed speed and radius. The same can be said for all the planets around the Sun. This gravitational force can describe the motion of every object in space and it is the case that every object that contains mass will exert gravity on every other massive object. Of course to feel the effects of gravity we must be near an enormous object, like a planet, but it is completely accurate to say that gravitational force is exerted by your car, your refrigerator, even you yourself. It is just that this force is completely negligible compared to the gravitational force exerted by the earth. Newton developed an equation to quantify the magnitude of the gravitational force between two objects, and it looks like this, where F is equal to the constant of universal gravitation, G, times the mass of the first object, times the mass of the second object, divided by the distance between them squared. This constant, like any other constant, simply exists so that a natural phenomenon like gravity can be expressed in our own arbitrary man-made units, and it is equal to 6.67 times 10 to the negative 11 Newton meters squared over kilogram squared. These are the units that will cancel out the units on the masses and radius so as to give a value for force in newtons. This value is not known to Newton at the time but was determined experimentally about a hundred years later by Henry Cavendish. When discussing the radius between two objects we will take the distance between their centers rather than their surfaces, as Newton showed that the gravitational force exerted by an object depends only on its mass and not on its volume, meaning that when discussing gravity we can treat everything as a point like mass. He had to invent the calculus to do so, much to the dismay of math students everywhere. When examining a system like the earth and the moon, we must understand that both of these objects exert gravitational force on the other, and that these forces are equal in magnitude meaning that both of these bodies rotate around their combined center of mass, but don't forget that F equals ma, so equal forces will not produce equal accelerations if the masses are different. As it happens, the earth is much much more massive than the moon so the mutual gravitational force is able to accelerate the moon more than the earth, and the center of mass for the system lies within the earth itself, which is why we simply observe the moon going around the earth. The same can be said for Newton's falling apple. The apple accelerates towards the earth and the earth accelerates towards the apple, but the earth is more massive than the apple by an inconceivable factor, so the acceleration of the earth is not even measurable whereas we can visually confirm the acceleration of the apple. Furthermore, we want to understand that an apple will fall to the earth with the same acceleration as a bowling ball or any other massive object, if we disregard wind resistance. Although counterintuitive to some, we can rationalize this if we understand that while the force of gravity is able to impart greater acceleration on a more massive object, the more massive object also has greater inertia or resistance to being accelerated, so the end result is that all objects accelerate towards Earth in the same way, at 9.8 meters per second squared. This fact is easy to derive if we do some algebraic manipulation. We know that a falling object exhibits behavior according to Newton's second law, F equals ma, where the force that generates the falling is equal to the mass of the object times its acceleration, but this force is the gravitational force, so we can also model the falling behavior with G m1 m2 over r squared, where m 1 is the mass of the object and m2 is the mass of the earth. If we set these equal to each other, the mass of the object is found on both sides and will cancel out, so we can see that the acceleration due to gravity is equal to the gravitational constant times the mass of the Earth divided by the radius squared. This means that the mass of an object does not affect the rate of free fall. Newton's work on gravity was revolutionary. It correlated an incredible amount of data, from terrestrial motion to celestial motion, which is all any good theory can hope to do. But he could not explain how objects can exert the gravitational force on one another from a distance. Later, scientists solved this problem by labeling gravity as a field force, stating that matter generates gravitational fields in space. This was a bit more satisfactory, but it wasn't until Einstein's general theory of relativity that we arrived at a more sophisticated understanding of gravity, which has helped us learn about the structure of space itself, as well as how planets and stars and galaxies form. We are still trying to fully understand gravity today, but the continuation of this discussion will have to wait until the modern physics course, so for now let's check comprehension. Thanks for watching, guys. Subscribe to my channel for more tutorials, support me on patreon so I can keep making content, and as always feel free to email me: by travelpod
