Gravity is the force that keeps us standing on Earth's surface. It's the reason that a ball thrown upwards falls back down towards the ground. It was Newton who first realized that this force, gravity, doesn't just affect physical objects here on earth, but is also responsible for the motion of the stars and planets. Gravity keeps the earth moving in orbit around the sun and the sun in orbit around the supermassive black hole at the center of the Milky Way galaxy. Gravity is a central principle in black hole physics because it's gravity that gives black holes their extreme properties. Until the year 1687, the year that Isaac Newton put forth his vision of gravity, no one had a clear understanding of what causes the attraction of objects towards the ground. A similarly mysterious force was also keeping the Earth moving around the sun. Even in antiquity, humans understood that something held objects in place, but lacked the mathematical description. It was Newton who provided the first empirical description of how gravity works. Although Newton was the first to explain gravity mathematically, almost exactly 100 years earlier in 1589, Galileo Galilei was busy investigating gravity and his observations greatly advanced our understanding of the interaction between objects and their masses. Galileo theorized that falling objects of different masses would fall at the same rate contrary to the Aristotelian belief that heavy objects fall faster than light objects. It's famously claimed that to prove this idea, Galileo climbed up the Leaning Tower of Pisa and dropped two cannon balls with different masses one heavier and lighter. He observed that if both cannonballs were dropped simultaneously they hit the ground at precisely the same time independent of their weights. Galileo made the mistake of assuming that the gravitational force was a constant between two objects with no relationship to the distance between them. Historians disagree whether this experiment really took place because it's first mentioned almost 65 years after it's supposedly took place in a biography of Galileo by Vincenzo Viviane. One experiment done during Apollo 15's mission to the moon demonstrates the principle that Galileo addressed. At the end of the last moon walk, astronaut David Scott performed the same demonstration that Galileo did with a hammer and a feather in the vacuum of space. The result of course is visible in this famous video. In my left hand I have a feather, in my right hand, a hammer. I guess one of the reasons we got here today was because of a gentleman named Galileo a long time ago who made a rather significant discovery about falling objects in gravity fields. And we thought that where would a better place to confirm his findings than on the moon. So we thought we'd try it here for you and the feather happens to be appropriately a falcon feather or a falcon and I'll drop the two of them here and hopefully they'll hit the ground at the same time. How about that? So, Mr. Galileo was correct in his findings. Shortly after Galileo's death, mathematician and astronomer Johannes Kepler observed that planets trace ellipses through the solar system as they orbit the sun. Kepler famously described the motion of the planets mathematically, laying the groundwork for the second last piece of the gravity puzzle which was solved by Christian Huygens, who in the 1660's described the law of centrifugal force. Together with the help of Edmund Halley, Christopher Wren and Robert Hooke, Isaac Newton had all the clues he needed to piece together the mathematical description of gravity. In 1687 Newton's book Philosophiae Naturalis Principia Mathematica which translate to the Mathematical Principles of Natural Philosophy. Newton laid the mathematical foundations to explain all of gravitationally related phenomenon including apples falling from trees and planets in orbit around stars. Gravity is an attractive force between two objects that have mass. Any object that we talk about in this course with the exception of light has mass. The earth has mass, I have mass, and you have mass. There's therefore a gravitational attraction between the earth and me, the earth and you, but also between you and I at any given time. The mathematical description of the force of gravity needs to take into account the mass of both objects, and also the distance between them. In order to get useful information out of any equation, we also need a universal gravitational constant to tell us how strong the force will be given the masses and the distances. Let's call the mass of the larger object capital M, and the mass of the smaller object little m. The distance between the two objects will be measured by a lowercase r and the universal gravitational constant will be denoted as a capital G. The force of attraction between two objects will be directly proportional to their masses, but inversely proportional to the square of the distances separating them. Direct proportionality means that the force F will be equal to the universal gravitational constant G times capital M times little m. Finally, because of the inverse-square relationship, we divide the whole right hand side of the equation by r to the power of two. This equation is called Newton's universal law of gravitation, and calculates the force between two objects no matter what their masses. In order to use this equation, we need to consider the units of each term. G, the universal gravitation constant has a value of 6.67 times 10 to the minus 11 in units of Newton meters squared per kilogram squared, and that's a mouthful. To make these units cancel out, you can see that capital M and little m will cancel out the kilograms squared term, and that the distance squared cancels out the meters squared term leaving behind Newtons which are a measurement of force. Notice how tiny the gravitational constant is. If we ask ourselves how much attractive forces felt between two objects each weighing one kilogram and separated by one meter, the answer of course is G times one kilogram, times one kilogram divided by one meter squared. So 6.67 times 10 to the minus 11 Newtons or 66.7 picoNewtons. For comparison 67 picoNewtons is about how hard you have to pull the two ends of a DNA molecule in order to have them unravel. But gravity acts on much larger scales and is therefore comparatively weak. Let's compare 66 picoNewton's to the force of gravity that I feel due to the Earth. Since Earth weighs 5.97 times 10 to the 24 kilograms and I weigh about 75 kilograms, in order to calculate the force of gravitational attraction, we'll replace capital M with Earth's mass and little m with my mass. We also need to know how far apart the center of the Earth is from the center of me. Let's take the radius of the Earth's surface to be r and replace it with a value of 6,378.1 kilometers, which we have to convert into meters. So, 6,378,100 meters, which we then square. Finally, we replaced the universal gravitation constant G with its value of 6.67 times 10 to the minus 11. And its units Newton meter squared divided by kilograms squared. Together, the units of meters cancel each other out as do the units of kilograms leaving Newtons in the result. I'll get my calculator out and plug in the math and I get the result of 735 Newtons. So, I'm being pulled towards the center of the Earth with a force of 735 Newtons. The unit of force Newtons is sometimes difficult to put into context. It's related classically with the acceleration of a mass by Newton's second law F equals ma, which relates the force on a mass to how quickly the mass accelerates. Since I feel the force of gravity as 735 Newtons, I can calculate my acceleration due to gravity by dividing my mass 75 kilograms which results in acceleration of 9.798 meters per second squared. You might recognize the coincidence. The acceleration I feel is very close to the value of Earth's acceleration due to gravity, which is often denoted as a little g, and has an average value of 9.807 meters per second squared. The reason that these two numbers are different is because the strength of Earth's gravity varies over a surface. For example, you weigh about half a percent heavier, when you're at the Earth's poles than you do when you're along its equator. In fact, Earth's gravity varies a lot over its surface because of the different densities of rocks and the different geography of regions. Earth's gravity diminishes by about one fifth of one percent from earth's surface to an altitude of five kilometers. So, your height above or below sea level is also a factor. But geology can account for another one 100th of a percent difference in gravity. This map of the globe represents the difference in Earth's gravity from the average value. Red indicates stronger gravity and blue indicates weaker gravity. The data was collected by a pair of satellites called GRACE, the Gravity Recovery and Climate Experiment. GRACE uses changes in Earth's gravity to measure changes to huge masses of ice in our polar regions. If gravity there decreases, scientists can determine how much of the glacial ice is melting in those regions and this data can even tell where vast underground reservoirs of water are filling up. If Newton had accomplished nothing but the mathematical formulation for the law of gravity, he would still go down as one of history's greatest physicists. But he contributed much more to our understanding of the universe. He revolutionized our understanding of motion, forces, and mechanics with his three laws of motion. Newton's three laws can be stated in the following way: Newton's first law states, an object at rest will stay at rest unless a force acts upon it. An object in motion especially uniform motion, will stay in motion unless a force acts upon it as well. It's interesting that we distinguish between an object at rest and an object moving with a uniform velocity. As we get deeper into this course, you'll understand that these two examples, an object at rest and an object in uniform motion are themselves within what we call an inertial frame of reference. We could also think about a rocket moving in outer space at a constant speed. Unless the rocket were to fire its thrusters to exert a force in the opposite direction, the rocket will continue moving at a constant speed forever. Newton's first law is also called the law of inertia. Inertia is the resistance an object has to changing its state of motion. Newton's second law states, an object acted upon by a force will experience an acceleration in proportion to its mass. This is the famous formulation which is described by the equation F equals ma that we used earlier. Any force acting on an object will produce an acceleration in proportion to the mass of the object. So, for any given force, a small mass will accelerate quickly but a large mass would accelerate slowly. Think about it using a small motor on both a small boat and a huge ship. The motor delivers the same amount of force but the ships accelerate at much different rates. Newton's third law states, for every action there's an equal and opposite reaction. Newton's third Law is a little hard to wrap your head around, but it basically means this, any force which is imparted on an object must also be imparted equally upon another. In other words, for all the force of Earth's gravity pulling upon me, I'm also exerting a force pushing down upon the Earth. This point confused me for some time as a student. Why is it that we say earth gravity has a value of 9.81 meters per second squared? That's a measure of acceleration. Well, when I'm standing still on Earth, Earth surface is not actually accelerating anywhere. My acceleration is zero. The truth of the matter is that the strength you exert to stand is the force pushing back on the Earth. The net force between you and the Earth ends up zero. Well, what about if you aren't standing on earth's surface but you've gone sky diving and you're falling freely through the air? In this case, you are accelerating at 9.81 meters per second squared but you should also consider that Earth is accelerating towards you. The forces are the same for you and for the Earth, but the acceleration of the two is different because you and the Earth have vastly different masses. In this case, Earth would accelerate towards you at a tiny rate of about 1.23 times 10 to the minus 22 meters per second squared. Newtons second law of motion means that when we apply a force to an object, the object will accelerate. Therefore, when you apply a gravitational force to an object, it will accelerate. If we take Newton's law of universal gravitation F is equal to GMm over R squared and set the force equal to F in Newton's second Law F equals ma, then the little m masses on both sides of the equations cancel each other out resulting in the equation a is equal to GM divided by r squared. This equation provides a simple way of calculating the acceleration due to gravity. When I stand on the surface of a planet that has a radius r and a mass capital M, then the acceleration due to gravity at the surface is simply given by G times big M divided by r squared. If the planet is earth, we use the symbol G to represent the acceleration due to gravity. We say that a body has a gravitational field when it has the potential to accelerate nearby objects towards it. Newton's equations are robust enough to send rockets to other planetary bodies. In order to do so, we need to further tie the concept of gravitational potential energy, the energy required to climb through a gravitational field in order to calculate escape velocity.
