 The falling of two objects in the Earth's gravitational field, at a constant rate of acceleration. The orbiting of the Moon around the Earth in regular cycles. The Earth pulls on the Moon, the Moon pulls on the Earth, but the tether is unseen. What connects these two phenomena, if anything? And if we can find the connection, what might it tell us about the deeper universe, the greater and grander distance scales at which matter and energy exist, and the clumping and collection of matter and energy over vast swaths of time into the visible universe that we know today. It is the hand of gravity that shapes all of these things, and we will begin a deeper exploration of gravity now. In this lecture, we're going to take a slightly deeper dive into a force, a deeper dive than we've taken into any other force. We're going to take a very close look at gravitation. And the reason that we're doing this is because gravity is recognized in the modern era as one of four fundamental forces that, when acting on their own or in concert together, shape what we understand as the everyday universe. The key ideas that we will explore in this section of the course are as follows. We will dig deeper into gravity as a force. We've taken a shallow look at it before. We've considered what it means to add energy into the configuration of an object near the surface of the earth or take that energy back and convert it, for instance, into kinetic energy. We've thought about falling in the presence of a constant acceleration due to gravity, but now we're going to understand where all of those things come from. To do that, we're going to need to learn the mathematics that describes the properties of gravity. But the good news is that we are merely going to be recycling the force, Newton's laws, and energy concepts that we have exercised already in the course. We will reconnect with the observations of gravity near the earth's surface as a source of constant acceleration and finally come to understand why that is. We will then think about how one can connect observations of motion in the presence of a gravitational force on the earth with the motion of objects in the larger cosmos that are interacting through the gravitational force. And finally, we will glimpse the nature of gravity, its true nature, its true cause in the universe. And then we'll take a look at some of the modern puzzles in gravity that if you were to pursue a career in physics, you might face as a physicist. We are really dealing here with one of the most ancient questions that humans have asked about the universe. In particular, the universe as they experience it here on earth. Why do things fall? It's very likely that since before recorded human history, our species has wondered continuously at why it is that things tend generally to fall when they're released near the surface of the earth. So pick up a rock, hold it out, let it go, it falls down. Why does the rock never rise into the sky? Now the earliest written speculations on the subject originate from places like ancient Greece or much later from what is now known as India, sort of circa 500 years into the common era. However, explanations incorporating both measurement and mathematical description originate really beginning with Galileo Galilei. There's a reason that he's often considered the first modern scientist. By combining careful observation of the natural world with careful experimentation to test claims about the natural world and rigorous mathematics to describe and predict the outcome of future experiments, Galileo represents a complete scientist in the sense that we think of it in the modern world. Now his careful experiments, they revealed that acceleration during falling is constant and it's independent of the mass of the falling object. Now this observation, carefully documented and well described, ran very counter to, for instance, ancient Greek claims, a la say Aristotle, that in fact the rate of falling depends on the mass. That just turned out to be experimentally completely false. And this is an excellent example of what it means to be a science. In science it's not enough to make claims. Anybody can make claims. What's important is how much evidence you have supporting your claim, the quality of that evidence and the ability of others to reproduce your evidence and verify or refute the claim. Galileo brought an extremely careful approach to this particular subject and in running very well controlled experiments where he could control the falling and take various uncertainties and variables out of this process, he was able to demonstrate something that people had only speculated about before pronouncing statements but never really backing up their claims with strong evidence. It would later be Johannes Kepler, who lived from 1571 to 1630, who would then assemble the most precise astronomical data of certainly his day and from this data infer three laws of planetary motion. Now at some point in a later lecture we will take a closer look at sort of specifically these laws, but basically the motion of celestial objects like Mars or Venus, Jupiter and so forth, these laws would apply to those objects as they move around the Sun. Now the reason for these laws was not at all clear to Kepler and their connection with for instance Galileo's observations of falling objects on Earth was also not clear. This was simply a set of rules that certainly seemed to apply to celestial objects but the fact that that had anything to do with falling on the surface of the Earth was not at all clear to Kepler and certainly not to Galileo. Ultimately it would be Isaac Newton who would unite in a sense Heaven and Earth by identifying a singular cause, a singular force that we now call gravity as the mutual cause both of falling on the surface of the Earth and the orbits of astronomical bodies like the moon around the Earth, the Earth around the Sun, the Mars around the Sun and so forth. In physics our community often considers this the first unification of previously disparate concepts under a singular framework. That is to say dropping objects near the surface of the Earth and the regular motions of planets in the night sky, you know even the regular motions of the moon around the Earth, these are all connected to the same fundamental cause and if you can understand that cause you can suddenly answer huge swaths of questions that nobody prior to this realization understood were even connected to one another. This trend of unifying previously disparate observations into a singular explanatory framework would actually continue through the 1800s with electricity and magnetism and also with the concept of matter and atoms to heat energy. Those unifications all kind of occurred in the 1800s and then into the early 1900s with thermodynamics, heat energy, the concept of the atom, motion, electricity and magnetism and this would all culminate in the 1960s in the unification of three of the four fundamental forces of nature, more on that later that we still are benefiting from today in terms of an era of scientific discovery and exploration but gravity still stands apart from this progress that was made in other aspects of unifying disparate phenomena into a singular framework although ironically it was the first unification that occurred in our understanding of nature. Let's begin to explore gravity as a force with its own associated law and the key concepts here are that you have a body, body number one with some mass M1 and you have a second body, body number two with some mass M2 and they are separated by some distance and what it was that Newton figured out is that regardless of what the cause of this force is there is due to the presence of for instance mass in these two bodies an attractive force between them and it was his illumination of the law that governs the masses, their separation and the force that results from that that still to this day is utilized continuously in our engineering of the world around us. Essentially quoting from Isaac Newton himself in 1666 I deduced that the forces which keep the planets and their orbs must be reciprocally as the squares of their distances from the centers about which they revolve and thereby compared the force requisite to keep the moon in her orb with the force of gravity at the surface of the earth and found them to answer pretty nearly. In other words what he found was that this treating this as a possibly two related phenomena planets moving in orbits but also things falling near the surface of the earth what Newton discovered was that if you attacked those two things assuming that there was a force that whose strength goes as the distance squared between the objects you could relate the moon going in orbit around the earth with the falling of objects in the surface of the earth with the same force equation. That's a stunning revelation. What is that equation? Well the force between object number one and object number two and quite specifically the gravitational force that object number two exerts on object number one can be denoted with this symbol here a vector f for force with a subscript one two which stands for the force that two exerts on one. This is related to a bunch of numbers. I'll come back to this constant g in a moment but not surprisingly it goes directly as the product of the masses of the two objects the bigger m1 the bigger the force of gravity between m1 and m2 the bigger m2 the bigger the force of gravity between object one and object two. So that's not a huge surprise. The key aspect of Newton's discovery here was that the strength of this force does not go as one over the separation but one over the separation squared. So in the denominator we have here the distance on a straight line between the two objects so for instance using this picture over here the mass number one is the blue dot mass number two is the red dot the straight line that connects them most directly is this distance r12 and the force goes as one over r12 squared. Now where does the force point? The force that two exerts on one is an attractive force so it must point from one to two and so the direction of this force is given by a unit vector whose body lies along the line connecting the two objects and that points from object one to object two if we're talking about the force that two exerts on one. Let's come back to this constant here the product of the mass is figuring that out that took time but wasn't in a sense the worst part figuring out that it goes as the distance squared that took time but again we know how to right distance in here but if you just take mass and you take the product of that you get kilogram squared and if you divide by the distance that's kilogram squared over meter squared that's not the units of force the units of force are what we now call Newton's those are kilogram meters per second squared so how does one relate m1 m2 over r12 squared to the force which must be in Newton's kilograms meters per second squared there must be some other number that multiplies m1 and m2 and the one over the distance squared and that number is known as Newton's gravitational constant g and it was determined to be a relatively tiny number g has a value of 6.67 times 10 to the minus 11 and the units on that are Newton meter squared per kilogram squared which makes sense this constant multiplied by the product of the masses over the the distance squared the meter squared cancel out on the bottom and the kilogram squared cancel out from the top and we're left with Newton's which is what we wanted in the first place again this is Newton's gravitational constant so far as we know its value doesn't vary with any of the things in this equation mass separation or even time for instance but of course you know these things are subject to revisitation as we continue to refine our understanding of the universe and it would be very interesting to find out that this constant is in fact not constant now m1 and m2 as I said are the masses under consideration that are acting on each other via gravity and r12 is the distance between the two masses and r hat12 is the unit vector that points from mass1 to mass2 if we're talking about the force that two exerts on one so this is the form of Newton's law of gravity or law of gravitation and it's not so bad we're gonna play around with it a little bit in this lecture video and of course you'll get to practice this on your own both in class and through homework so let's as an example of this think about the magnitude of the gravitational attraction between the sun and the earth the earth pulls on the sun pulls on the earth we normally think of the earth as going around the sun and the sun is holding us in orbit but in fact the earth is pulling back on the sun and we're going to explore this a little bit now and think about the consequences of all this so the earth and the sun in fact mutually exert gravitational forces on each other let's start by looking at the force that the sun exerts on the earth so we have from the previous view graph here we have the force of two on one two in this case will be the sun one will be the earth and so we can write down Newton's law of gravity for the force between the earth and the sun where we're considering the sun acting on the earth so we have Newton's constant that never changes we have M1 the mass of the earth M sun the mass of the sun and then we divide by the distance squared between the earth and the sun now we can actually plug numbers into this equation we can go look this up on Wikipedia or perhaps look it up in the back of a textbook the mass of the sun is 1.99 times 10 to the 30 kg it's extremely heavy the mass of the earth is much smaller but still big 5.92 times 10 to the 24 kg and the distance of closest approach between the earth and sun system because the earth does not make a perfectly circular orbit makes something called an ellipse so a slightly elongated orbit and that means sometimes we're closer to the sun and sometimes we are further from the sun the distance of closest approach between the earth and sun system is about 147 billion meters or about 147 million kilometers so plugging all these numbers in we find out that the force that the sun exerts on the earth is 3.66 times 10 to the 22 newtons now what direction does it point in? well the sun is pulling on the earth so this force must point from the earth toward the sun and I've indicated that with an arrow here the unit vector r hat earth, sun would point from the earth toward the sun but Newton's third law states that if the sun exerts a gravitational force on the earth the earth also exerts a gravitational force on the sun so what is this force? again we can just use the law of gravity we can find its magnitude the force that the earth exerts on the sun is just newtons constant times the mass of the sun times the mass of the earth divided by the distance between the two of them squared now here so far nothing has changed we still have the mass of the sun times the mass of the earth in the numerator we still have the distance between them squared in the denominator and we still have newtons constant nothing in the magnitude equation above changes these numbers are all the same so we find that the magnitude of this force is again 3.66 times 10 to the 22 newtons but the direction has now flipped when we're talking about the force that the earth exerts on the sun that is an attractive force that pulls the sun toward the earth it starts from the sun it points to the earth so each force has the same magnitude but mutually opposing directions so why is it that the earth appears to do most of the moving in the solar system well, think about it for a moment the earth has a much smaller mass than the sun so while the sun exerts the same force on the earth that the earth exerts on the sun the earth has much less inertia much less resistance to changes in its state of motion than does the sun so if one considers the acceleration that the earth would experience when pulled on by the sun it's much greater than the acceleration that's experienced by the sun as it's tugged on by the earth with the same force for the same force but different masses the accelerations will be different according to newton's second law of motion and so this helps us to understand a little bit why it is that the less massive of these two objects is going to tend to do most of the actual moving now let's revisit something that we looked at earlier in the course in the context of falling near the surface of the earth we looked at from the perspective of potential energy and changes in potential energy in the presence of the gravitational force of the earth near the surface of the earth it bears on one of the great questions that Galileo himself could not answer he could measure it, he could observe it he could describe it but he could not explain it newton however did the question that Galileo couldn't resolve in his lifetime but that newton in his own lifetime by discovering not only the laws of motion but the law of gravitation was the following why is it that two objects with radically different masses when dropped from the same height above the surface of the earth accelerate toward the surface of the earth at the same rate 9.81 meters per second squared that, while an observation of Galileo's was not explained by Galileo but newton armed with his laws of motion and the law of gravity figured it out so we can consider an object of mass m dropped from a height h above the surface of the earth and we can then try to use newton's second law and the law of gravity to figure out what's going on with this acceleration why is it that the mass of the dropped object doesn't affect any of this even though mass appears in newton's second law and mass appears in the law of gravity well, we can relate the two we know that the force of gravity on the little mass, little m, due to the earth is given by newton's law of gravitation the constant g times the mass of the earth times the mass of the dropped object divided by the radius of the earth plus the height all squared now what's going on in the denominator here remember the center of mass concept an object like the earth of mass m so mass of the earth in this case will act on another object as if all of its mass is concentrated at its center of mass so if you need to know what point you are supposed to consider the force of gravity being acting outward from in this case, it's the center of the earth assuming that the earth is uniform density, at least as a function of angles around the center of the earth now that's not exactly true the density of the earth is not the same everywhere and so forth but to this approximation assuming an average density for earth it's the same, no matter where you're standing on the surface of the earth looking down to the center we can assume that the center of mass of the earth is located at the center of the earth and gravity will act as if all the mass is concentrated at that point the center of mass of the earth and pulling on the second mass, little m now, we have the form for the force we have an object to height h above the surface of the earth but we're also a distance, the radius of the earth from the center of mass of the earth so this is the distance we have to put down here in the denominator we've got this all worked out this must be equal to mass times acceleration Newton's second law of motion so finally we can write an equation using the law of gravity for the acceleration due to gravity at any height h above the surface of the earth and what we observe here is that the mass of the falling object appears on both sides of this equation law of gravity equals mA m drops out of both sides and so we arrive at the equation for the acceleration due to gravity at any height h above the surface of the earth it's just Newton's gravitational constant times the mass of the earth divided by the radius of the earth plus the height above it from which you're dropping the object that whole thing squared we can define this as what we've been calling g but what we notice here is that g, which we have been taking as 9.81 meters per second squared and treating it like it was a constant at any height h above the surface of the earth it's not actually a constant it varies with your distance from the surface of the earth, the height h so technically g has a different value one meter above the surface of the earth than it does ten meters above the surface of the earth than it does a hundred meters above the surface of the earth so why don't we worry about that? well let's calculate the value of g for a few standard heights above the surface of the earth now what I'm going to do is I'm going to use the average radius of the earth to be about 6.3781 times ten to the sixth meters and I'm going to use for instance a first dropping height of one and a half meters above the ground that's roughly dropping from about shoulder height okay so if I punch all the numbers in I put in the Newton's gravitational constant I put in the mass of the earth I put in the average radius of the earth and I put in this height, 1.5 meters which is a tiny addendum to the radius of the earth I find out that I get an acceleration predicted of about 9.798 meters per second squared well that's not bad that's wildly close to the accepted average value of 9.81 meters per second squared great okay so nothing seems too crazy so far what if instead we go up a kilometer and drop an object so now I'm not at one and a half meters above the surface of the earth I'm a thousand meters above the surface of the earth great so I go up a thousand meters and I drop an object what acceleration due to gravity will it experience starting at that height of a kilometer above the surface of the earth well punch in the numbers grind through the calculation and you'll find that the acceleration due to gravity at this height, h2 is 9.7949 meters per second squared so it differs in the third decimal place from the value about a meter and a half above the ground and you can really begin to see here that g doesn't change too quickly as a function of height and why is that? it's because the radius of the earth absolutely dominates in this equation up here yeah you're adding a little number on to the radius of the earth the radius of the earth is huge and so as a result of this you're adding a tiny number to a big number it's not really changing this value of the acceleration due to gravity all that quickly it's changing but going up a hundred meters or so only caused it to change in the third decimal place so you can see why it is that we we wave our hands and say well just assume an average constant acceleration to the surface of the earth to be honest for the kinds of measurement instruments the Galileo would have had in his day it's not clear to me that he would have even been able to detect a change like this that substantially alright let's go up even higher let's go to the altitude of the International Space Station so the ISS is located at an altitude of 435,000 meters above the surface of the earth 435 kilometers at its greatest value that's the furthest distance you'll tend to find the ISS on the surface of the earth so plug in the numbers and calculate and now you'll see we're getting some movement here the gravitational acceleration due to the earth pulling on the International Space Station with its gravitational force is actually 8.5 8.6 meters per second squared that's about 10% lower or so than on the surface of the earth or 100 meters above the surface of the earth and so is it that people float around on the space station well remember your uniform circular motion the ISS orbits the earth every 90 minutes or so it's roughly speaking going in a circle around the center of the earth and so yeah it's true that the ISS is falling at a rate of 8.587 meters per second squared but it's also moving further away from the earth on a line that's tangent to its circular path and so it continues to circle and circle and circle so the ISS is in fact falling toward the surface of the earth it just keeps missing because it's also moving horizontally so that's the secret to flying that's the joke that Douglas Adams wrote into his series of books The Hitchhiker's Guide to the Galaxy and the books that followed on from that what's the secret to flight you throw yourself at the ground and miss the ISS has been throwing itself at the ground and missing ever since it was put into orbit and it's moving toward the earth the astronauts inside of it experience essentially microgravity they're all falling together toward the surface of the earth and so they're essentially in freefall and in freefall they don't notice that there's a gravitational acceleration so the key thing here is that for the acceleration due to gravity the key factor is the mass of the earth and the distance from its surface it's not the mass of the dropped object so nowhere in this exercise certainly G can change as a function of height above the surface of the planet earth in this case but the mass of the dropped object never matters in any of this so what's amazing about all of this is that the law of gravity acts without regard to whether you are on earth or elsewhere in the universe and that is ultimately what it means to be a law gravity is the same everywhere in the universe now that's a hypothesis that one has to test well how would you test it? okay, observe other massive bodies moving around each other look at how they're accelerating are those accelerations consistent with the law of gravity and in basically every case that we have ever been able to test this in it holds and so as far as we know this is a universal force and that's what makes it one of the four fundamental forces of nature gravity acts without regard to where you are on earth or anywhere else in the universe it explains falling on the surface of the earth it explains the orbit of the moon around the earth it explains the earth going around the sun it explains the orbits of distant binary stars stars that co-orbit each other it explains a lot of things why we live in a cluster of stars known as a galaxy and so forth gravity shapes all of that so ultimately Newton who apocryphally was sitting under an apple tree watching an apple fall when he had his sort of key inspirational moment as to what might be going on here with falling and orbiting it doesn't matter whether we're talking about stuff on earth or stuff up in the sky and so indeed this rule of nature has helped us to understand a whole bunch of things that were a mystery even Galileo wrote a great treatise on the tides and got it all wrong because he didn't understand the law of gravity he just didn't have that insight that Newton had and then the math to work it out so the law of gravity has helped us to understand why ocean bodies of water move why is it that there's a low tide when the ocean appears to recede away from the shore why is there a high tide when the ocean seems to move in over the shore and go further than it normally does on average it's because the moon's gravity pulls on the oceans and the oceans are sloshing around on the surface of the earth tugged by the gravity of the moon and very grossly speaking this is why water moves away from the shore as the moon pulls it away and when it moves back over the shore as the moon swings around the earth and pulls on it in the other direction it's helped us to understand the orbits of planets including the earth itself around a parent star we understand very well now why Mars and Venus and Mercury and Jupiter and Saturn all have the motions that they do in the sky and gravity is the great unifier gravity is the great unifier so falling on earth and being a planet going around a star have a lot in common and the fact that our solar system participates in a much larger collection of stars known as the Milky Way Galaxy that is explained by the fact that the gravitational attraction of all the stars on one another helps to organize them into large scale structures whose motions are all connected with one another due to the stretching out of gravity over vast distances look at the law of gravity there's no cutoff in distance is pulling on a star that is billions of light years away it's just so small we don't notice it and the star doesn't notice it there are many more important effects that are close by that matter but in something the size of the Milky Way there's a lot of pulling going on from star to star and also from things that aren't stars like hot gas that also makes up the Milky Way these cloudy regions in the band of bright stars that form the Milky Way in the night sky that's dust that blocks the light from the stars and the core of the Milky Way from reaching your eye those aren't places where there are no stars those are places where dust obscures our view of even more distant stars and so dust and gas play a big role in the gravitational binding of a galaxy but also other things that we can't see dust can be seen in different wavelengths of light wavelengths that for instance our eyes may not be sensitive to but they're also transparent in other wavelengths of light and so that's how we can see through dust light plays a big role in our understanding of the structure of things like galaxies but the structure of galaxies themselves and clusters of galaxies and clusters of clusters of galaxies these are structures that span the visible cosmos those structures can't be explained unless there's other kinds of matter out there that neither emit light nor absorb it these are vast structures and they are controlled by gravity but they are not dominated by the kinds of matter that make up you and me or the hot gas or the stars in our galaxy there's a dark matter that's out there that appears to greatly shape the structure of the cosmos and we're only beginning to get a first glimpse of understanding how that all works but it's thanks to gravity so far that we have any insight at all gravity is the great cosmic hand that affects all things in the cosmos but only so long as other forces like electricity and magnetism or electromagnetism don't overwhelm it gravity is actually the weakest of the four fundamental forces of nature that we know about the reason we notice its effects is because planets are big things and we are small things compared to planets and so planets get a big say in what happens to us as we move around on their surface but think about it this way electromagnetism allows atoms to bond or repel from one another so the fact that gravity might as it is doesn't pull you through the surface of the earth as thanks to the repulsive force between electrons due to electromagnetism and so gravity is weak because you don't fall through the earth if gravity were stronger than electromagnetism you would not be here to be listening to this lecture in the first place so there are other forces and over short distances especially when the masses get very tiny gravity plays very little role in what happens at those distances electromagnetism the weak nuclear force the strong nuclear force I'll comment more on those later they can play much greater roles over short distances and totally wipe out your ability to know what gravity would be doing at those same distances now let's revisit energy in light of the law of gravitation and in particular let's go back and revisit our old concepts about gravitational potential energy recall that in general for a conservative force F it has an associated potential energy U and the relationship between U and F is given by taking the derivative with respect to space of the potential energy that is looking at how the potential energy changes as a function of position in space the change of U with respect to position is related to the force that's exerted by that conservative potential so this operation this action of a derivative with respect to space on the potential energy is known as the gradient operator and all it does is tell us how potential energy changes with coordinate location that in turn by multiplying by a minus sign gives us the force as a function of coordinates the force of gravity changes only in the radial direction it gets weaker as you move away from the center of mass of the earth it gets stronger as you move along a line closer to the center of mass of the earth if you move perpendicular to the radial line that connects U and the center of mass of the earth the force of gravity doesn't change it only changes if you get further away along a radial line or get closer along a radial line but if you move at the same distance but perpendicular to the radius of the earth the force of gravity doesn't get any weaker or any stronger so all we have to concern ourselves with are changes in gravitational potential energy along the radial direction so how would we look at that well we have the law of gravity that gives us the force due to gravity between any two massive objects we want to find out what the potential energy is associated with the force described in that law well the force of gravity must be equal to the negative of the spatial derivative with respect to radial position of that potential energy function u of r, whatever it is well we can invert this equation by taking an integral of both sides with respect to d r and we can figure out exactly what it is that the potential looks like so the potential energy as a function of radius for gravity will ultimately be equal to the negative of the integral of the force of gravity d r well so plug it in you've got g m1 m2 over r squared take the integral of that with respect to r you'll find that the solution to this integral is negative g m1 m2 over r the distance between mass one and two plus a constant of integration which I'm labeling u0 well being a constant of integration in principle we can set it to be anything we want now it may have better values than others in some physical problems but for the rest of this discussion I'm going to set this constant of integration to zero and in that case the potential energy at any distance r from the center of mass for instance of m1 acting on m2 would be given by this equation negative g m1 m2 over r so let's go ahead and write that simplified equation the potential energy due to the gravitational force at any distance r is given by negative g m1 m2 over r one two all right again that's just the distance between mass one and two this is in fact the formal definition of gravitational potential energy we have the law of gravity we did this little calculus exercise we figured out the formula for the potential energy it gives us the potential energy any distance from the center of mass of an object so how do we make this square with our old friend potential energies m times g the acceleration due to gravity times h the height above the surface of the earth these equations don't look anything like one another so what's going on here okay well let's see if we can figure it out let's begin by defining m2 as the mass of the earth all right so m2 appears to be the mass of the earth and m2 is going to be acting on m1 so m1 is going to be the mass of some object that's going to be subject to the earth's gravitational force a ball in my hand for instance that I hold out now what is h in the equation mgh well that height h was always referred to specifically when talking about the gravitational potential energy for instance with respect to the surface of the earth that is a height h above the surface of the earth not a distance h from the center of mass of the earth which is what matters in the law of gravity but a height h above the surface of the earth so with that in mind we can take our general potential energy equation and we can construct a formal definition for what the potential energy relative to the surface of the earth would have to be it would have to be this it would be the difference between the potential energy a height h above the surface of the earth which is at a radius of the r earth the radius of the earth minus the potential energy as measured right at the surface of the earth okay well I can plug in with the above equation for potential energy substituting for m2 for the mass of the earth and m1 for the mass of the object to be dropped if you do a little algebra and pull out all the common constant factors that multiply the terms in this subtraction here you will find that this is equal to g times the mass of the earth times the mass of the dropped object times this thing which looks pretty awful at first it's negative one over the radius of the earth plus one over the radius of the earth well we can simplify this one step further and you'll see why I'm doing this in a moment by taking r earth out of the denominator of both of these fractions okay so if I pull one over r earth out of this parenthetical expression then I wind up with a term left over in the parentheses that looks like this 1 minus 1 over 1 plus h over r earth so now I have this number h over r earth in the denominator of this beast inside of the parentheses how has this made this any simpler well this comes from experience with some mathematics and I'm going to show you what I've learned to do in situations like this and why I did this in the first place by considering some essential ingredients in calculus and functions that will show us why I did this and where to go from here so here's a useful mathematical aside and as I said this is an aside on functions and series so as you might have learned from a pre-calculus course or a calculus course functions of variables like f of x can often be written in terms of a series of terms involving the argument x and these series traded for the function can actually make the functions easier to handle in certain algebraic situations so for instance you might have a nasty function of x but if you write out its series representation do a bunch of algebra with it that you needed to do with f of x in the first place and then at the end see what you've got left you'll often find that this although it seems complex at first actually makes your problem a lot easier to solve and in fact that's going to be the case here so just as an example of this let's imagine we have the following function 1 over the square root of 1 plus x this can also be written as 1 plus x raised to the negative 1 half power so these are equivalent to each other this is not the most pleasant function I mean you probably run into a function like this before and in a complex algebra manipulation situation this might be horrible to have to deal with you wind up squaring both sides of an equation and then a bunch of other horrible things happen very unpleasant however in the special case that that number x whatever it is has a value that's much much smaller than the number 1 this function can actually be represented by a series of terms involving x and other numbers so for instance here's the series this function 1 plus x all raised to the negative 1 half power can actually be written as the following 1 minus 1 half x plus 3 eighths x squared minus 16th x cubed plus a bunch of other terms in fact in principle an infinite number of them but if x is sufficiently small compared to 1 you don't always need to keep all the terms to keep going to solve the problem keeping just a few of them might be sufficient to substitute for the original function and then make some algebraic progress let's take a look at what I mean by this so for example here's the series representation of this function 1 plus x to the negative 1 half power so let's try putting a small number in for x and see what happens the good news is is that the function on the left side of this equation is really easy to punch into a calculator so it's very instructive to do this and then repeat that exercise with the series on the right hand side and see what kinds of numbers you get out for values of this function at certain values of x so for instance if x is 0.01 and I plug that in to the exact function I find out that the left side of this equation the exact function yields a number 0.99504 now I'm only keeping this to five decimal places of precision I could have gone further but this will already be sufficient to illustrate the point I'm trying to make now do the same thing plug in x equals 0.01 on the right hand side but just using the four terms that I have shown you here there are more terms in this expansion but just try plugging this into your calculator or some other computer program and solve for the value of the right hand side of this equation using just these first four terms in the series and if you do that you'll find out that you get 0.99504 so to the same level of precision as I kept the left hand side keeping just the first four terms as shown above gives exactly the same numerical answer to this level of precision that is fully accurate at this level of precision now what if I just keep the first three terms one minus one half x plus three eighths x squared okay plug it in plug in x equals 0.01 and you'll find out that again to these five decimal places it's 0.99504 the same number as the exact answer at this same level of precision so at this level of precision it's fully accurate we didn't even need to keep the first four we could have thrown out four and all the rest terms and still have gotten the same number to this level of precision let's just keep going let's just keep two terms plug in x equals 0.01 and this time you'll start to see some cracks opening up in this calculation now the right hand side just using one minus a half x gives us 0.9950 so if we wanted to be fully accurate at a precision of five decimal places this approximation probably wouldn't work you see that we have 99504 versus 99500 we're off in this last decimal place maybe that is actually sufficient for the problem you're trying to solve but for the purposes of this illustration I would consider that unacceptable we're keeping too few terms to get the target five decimal accuracy that we were looking for but what's the lesson from this when approximating a function by a limited number of terms in a series expansion you have to be careful about keeping enough terms to get the desired accuracy but if you do that if you are careful and you think carefully about it and then chop off all the terms you don't need you can trade the function for an approximate series representation now I want to just comment here at the end for those of you that have some experience with this that the above series representation of one plus x to the negative one half power is done using a Taylor series approach now that's something you would learn in a calculus class for instance now let's return to our problem our problem does not involve one over the square root of one plus x our problem involves one over one plus x where x is h over the radius of the earth a very small number so what we want to do is we want to find a reasonable series representation of this function one over one plus x that will give us enough precision that we can just plug it in and see if we can whittle down, simplify this equation further so in this case we are going to use a Laurent series and the Laurent series representation of one over one plus x is one minus x plus x squared minus x cubed, etc go ahead, try it out pick x equals 0.01 plug it into the right hand side see if you get the same number of terms you'd have to keep in order to get some reasonable level of precision two decimal places, three decimal places, four decimal places something like that now the number we're dealing with is not 0.01 it's h over the radius of the earth the radius of the earth is about 6500 kilometers and the heights we're talking about here are a few meters maybe a hundred meters 200 meters, a kilometer something like that is smaller than one so we absolutely can use this Laurent series expansion because x is much much less than one and in fact x is a lot smaller than the numbers that we have been dealing with in the examples I've been giving you in the last couple of slides here so I'm going to argue that all we need to do is trade the actual function for one minus x now that seems nuts but try it out see what adding in the h squared over r squared term would add to this and you'll see that it's affecting things in somewhere like the fifth or sixth decimal place which for the purposes of getting potential energy near the surface of the earth is probably not necessary we don't need one in a million precision for this exercise one in a thousand is probably good enough already so let's employ this series and an approximation using this series to help us continue the study of gravitational potential energy near the surface of the earth okay so let me remind you what the formula was we had calculated the potential energy at a distance from the center of the earth of r earth plus h and we subtracted from that the potential energy just at the surface of the earth at the radius of the earth this was equal to Newton's gravitational constant times the mass of the earth times the mass of the object we're holding at that height h times this thing in parentheses that looked horrible what do we have over here 1 over 1 plus x x is a number much smaller than 1 so we can immediately use our series expansion where x is equal to h over r sub earth and I'm only going to keep the first two terms in the series 1 minus x so plugging that into this equation I wind up with 1 minus 1 minus x where x is h over r earth well look at that what do we see 1 minus 1 gives us 0 these 1's cancel each other out and then we're left with a positive number h over r earth so let's go ahead and take advantage of these little simplifications that have occurred I've got Newton's gravitational constant times the mass of the earth over the radius of the earth squared times the mass of the object I'm holding at a height h times the height above the surface of the earth that I'm holding it and what is this thing? gm earth over r earth squared why it's just the gravitational acceleration of an object above the surface of the earth so we have recovered that the potential energy of an object of the height h above the surface of the earth is just the acceleration due to gravity times the mass of the object times the height above the surface of the earth where we are holding it we have recovered the gravitational potential energy equation for objects at typical modest heights above the surface of the earth this is incredible the fact that we don't have to any longer assume that g is constant we can take into account the fact that it changes with altitude and yet we can still do potential energy calculations is remarkable after all using energy to solve difficult problems is often the preferred methodology and thanks to the law of gravity that gives us the force of gravity between two objects of mass m1 and m2 separated by distance r we can now exactly write an equation for the potential energy at any distance from between the objects in this case we've looked at what it would mean to look at the potential energy of an object a height h above the surface of the earth and we've just by making a reasonable approximation with this series expansion recovered exactly the old formula for potential energy of an object of mass m a height h above the surface of the earth now let's look at a new idea it's great that we can revisit an old idea and reconcile any at first perceived differences between new and old so let's take the new idea and take it someplace we haven't gone before and that is thinking about how to get an object away from a planet so let's imagine we want to launch a rocket into space and have it go, keep going go go go go go, we want it to get away from the gravitational pull of the earth and go to other places, go to Mars go to the moon so what we're going to do is we're going to take a look at gravity and specifically for instance the gravity force that the earth would exert on something like a rocket with its own mass and think about what kinds of energy are required to get that object to escape the earth's gravitational force so we're going to take a look at gravity but this time we're going to think about the minimum speed at which you would have to launch an object straight off the surface of the earth along a radial line outward from the center of the earth in order to make it escape with the influence of earth this minimum velocity is known as escape velocity and we can calculate it using a simple thought experiment kinetic energy and potential energy so our initial situation is we have a vehicle like a rocket and we're going to give it a speed v0 it has a mass m and it's starting on or very near the surface of the earth so basically it's starting at a distance from the center of mass of the earth we're going to assume that we accelerate it very quickly up to whatever the speed is now it's moving radially outward away from the center of the earth obviously this rocket is beginning with some kinetic energy kI but it's also got some gravitational potential energy uI it is feeling the pull of the whole earth from its center of mass we can't ignore that that's the thing that is essentially keeping our rocket from getting away in the first place now we want to escape to escape it means we need to find a place in the universe where the potential energy between the rocket and the earth at that distance is zero so stare at this equation for a second and think about ideally where would you have to go in the universe in order to make the potential energy between the earth and the rocket be zero well g doesn't change it's a constant and the masses of the objects are not changing in this problem the mass of the earth is the mass of the earth the mass of the rocket is the mass of the rocket but the distance between them is changing and how far away would I have to move this rocket in order to get the potential energy to be zero well I would have to move it to infinity if I set r12 to infinity I get 1 over infinity and 1 over infinity is zero so the potential energy between the rocket and the earth is zero when the distance between them is infinite now that sounds nuts but maybe going to Mars is far enough away for this to be true or maybe going half way to the moon is far enough for this to be good enough true enough for our problem infinity is a big number but the universe is also very large so we probably don't have to go to infinity to understand what's required to do this but you know for the purposes of this mathematical exercise we're going to exactly treat this limit that the potential energy due to the gravitational attraction between these two objects will be zero when the distance between them is infinite okay so let's imagine we achieve a velocity v0 that can get us to an infinite distance from earth but we want to know the minimum velocity required to do this overkill but what's the minimum required well the minimum required would be the velocity such that at the end of the journey when we reach infinity we have no velocity anymore with respect to the earth at that point our kinetic energy would also be zero the potential energy would be zero the kinetic energy would be zero and so at the end of this grand journey we're imagining here the kinetic and potential energy is zero let's write down our conservation of energy equation the total energy present at the beginning when the rocket is leaving away from the surface of the earth at a speed v not it has kinetic energy it's got some gravitational potential energy and then it flies off to infinity and it comes to a stop at that point it has no kinetic energy and no potential energy due to its attraction to the earth so we have a situation where we have m v not squared that's kinetic energy we have minus g m times the mass of the earth divided by the radius of the earth that's the gravitational potential energy that the rocket would experience that whole thing has to be equal to zero because after all at infinity we have no speed and no potential energy due to gravity awesome this isn't so bad we can just rearrange this equation and solve for v not and so for instance if you do that you'll find that v not squared is equal to 2 times the gravitational constant times the mass of the earth divided by the radius of the earth in fact to find the escape velocity for any body with mass m and surface radius r this is the equation you need so whether it's an asteroid and the asteroid belt like Ceres or the planet Mars or the edge of the atmosphere of Jupiter if you know these numbers you can calculate the escape velocity required to get an object basically completely away from the gravitational influence of the parent mass like Jupiter or Ceres or earth so if we plug in the numbers for earth the minimum speed that we would need to achieve to completely escape earth's gravitational force is 11,000 meters per second or 40,000 kilometers an hour or 25,000 miles per hour that's a big number but it's about right so you can do a deep dive on escape velocities and rocketry if you like but this gets the basics of it this captures the basics of it immediately if you want to get an object let's say into orbit around the earth or beyond the orbit of earth so maybe moving into interplanetary space on a journey to Jupiter or Saturn you basically need to get that object up to at least this speed that's what it's all about so if you want to successfully escape earth's gravity so that this thing can move freely in outer space without being yanked back down to earth when it runs out of fuel now we've been talking about for instance a rocket subject to the gravitational force of one thing like planet earth but the universe isn't full of just one rocket and one planet earth there's earth, there's the rocket there's asteroids in the asteroid belt there's the objects in the orc cloud there's other stars the universe is a complicated place so how do you deal with more than one gravitating body how do you figure out the force say on a space probe when the moon is pulling on it and the earth is pulling on it now you have two things now acting on the probe so what's the net result what's the net force that results on that probe and you can extend this example up to any number of bodies that can exert gravitational forces on each other so we have a probe that is at the moment shown in this picture on a line that connects the earth and the moon so it lies somewhere between the earth and the moon it's a distance r1 from the earth and a distance r2 from the moon and when I say that I mean the center of mass of the earth and the center of mass of the moon so what's the total gravitational force on this probe and as a bonus what's the total gravitational potential energy that this probe is experiencing at this moment well to find the total force we can simply employ what is known as the principle of superposition forces add like vectors if I have two forces acting on the same object I add those forces as vectors and I can get the total resulting force so this shouldn't be alien this was essentially what we were talking about in Newton's first and second laws the sum of the forces equals mass times acceleration well if you want to know the total force sum the forces as vectors and nothing is different here so the total force on the space probe will be equal to the vector sum of the forces between the earth probe system and the moon probe system the earth pulling on the probe is one force the moon pulling on the probe is a second force add them together as vectors okay let's put this into equations so the total force on the probe will be the sum of the individual probe and object pairwise forces so I is a sum that goes from one to n here we have n equals two two bodies i equals one is the earth i equals two is the moon so I can write that sum explicitly I only have two terms in it so I have the force that the earth exerts on the probe and I have the force that the moon exerts on the probe and I add those two together let's keep going so again I have these totals let's assume that all the action happens along that straight line that connects the earth and the moon in fact it will in this case because I intentionally put the probe on that line so I have a line connecting the earth the probe and the moon one line three points let's just call this the x axis so henceforth I will label everything that happens as if it's happening on the x axis of a coordinate system so I can fill in the above equation the sum of the forces using the law of gravity and write each force vector now remember that the direction of force of the force that object i exerts on the probe is in the direction of a vector pointing from the probe toward i so what you should find if you try this on your own with the picture I just showed you is that the total force will be equal to the sum of two terms the force that the probe feels due to earth's gravitational pull so that's gm earth m probe over r1 squared in the direction negative i hat pointing from right to left from the probe back to the left toward the earth and another force g times the mass of the moon times the mass of the probe divided by the distance between the probe and the moon squared so that's r2 in a direction plus i hat pointing from the probe to the right toward the moon so these forces act in opposite directions their individual magnitudes are determined by these terms out in front of the unit vectors the directions are determined by the unit vectors just like all other forces that we have dealt with so I can pull the common unit vector out in front I can pull common constants out in front and I'll be left with an equation that looks like this g times the mass of the probe in parentheses times i hat gives me the total force so the magnitude of this force will be the gravitational constant times the mass of the probe times this term in parentheses and the direction will be determined by i hat and any lingering signs that are left over when I do this subtraction now if I'm the moon is less massive than the earth so in order for the moon to win out in this tug of war I have to be in a place so that the mass of the moon divided by the distance to the moon is bigger than the mass of the earth divided by the distance to the earth so as I move closer to the moon the moon wins out in this little tug of war and will tend to accelerate me toward it but if I move closer to the earth eventually it wins out in this tug of war and instead I will be accelerated toward the earth and there's a place somewhere in between the two where this exactly cancels you can solve for that if you want to and then there's no net force this is a point of equilibrium where the net force of the moon pulling on the probe and the earth pulling on the probe is exactly zero so already you can start to think about things like where is the equilibrium point how would I solve for that where would I have to be if I wanted the moon to win this tug of war where would I have to be if I wanted the earth to win this tug of war already a lot of things start flowing from this equation imagine if you have three objects four objects all pulling on the probe these are the kinds of things that actual rocket scientists have to take into account when they're thinking about missions to you know places like the moon which are very close to other gravitating bodies of mass like the earth okay so as I said the magnitude of the force is given by this mathematical term here and we have the earth and the moon system playing a tug of war on the probe pulling in different directions so what is the total gravitational potential energy of the probe depending on its distance of course from these two bodies now to answer this let's remember the relationship again between force and potential energy for a conservative force the force is the negative of the gradient of potential energy well what's the force on the probe well the force on the probe in this case is the sum of these two forces well well if I then consider these individually to be the negative of the gradient of the potential energy between the probe earth system and the potential energy between the moon probe system I will find out that in fact I can write this total force as the negative of the gradient of the sum of the potentials of these two individual pairwise systems so in order to get the total potential energy involved in this situation you just have to calculate the individual pairwise potential energies and add them up this skill by the way will be universally applicable to any and all conservative forces the electric force is a conservative force so get ready to do this all over again in second semester physics it's the same principles F it was the negative of the gradient of U I can sum the forces to get the total force I can sum the pairwise potentials to get the total potential energy this will come in immensely handy when thinking about the energy of a configuration of ions charges, atoms that are acting on each other through for instance electrical forces it will be very similar to dealing with bodies that can act on each other through gravity and the potential energy of their configurations the mathematics is identical it's just for a different force so what do we get the total potential energy of the probe earth moon system here from the perspective of the probe at least will be the potential energy due to the probe earth configuration and the probe moon configuration added together well I can immediately substitute with the potential energy equations for these two systems and then I can pull out common numbers and factors and so forth and I'll find that this is the equation that describes this particular situation that we're considering so bad we see that when dealing with conservative forces like gravity we have a lot of information at our fingertips if we remember how to relate the force and energy concepts we also are reminded again why energy is so nice to deal with if I hadn't been kind and put the probe along a line connecting the earth and the moon then we'd be dealing with x components and y components and possibly z components that's a lot of vectors for the forces but for the potential energies all that matters are the straight line distances from the probe to the earth and the probe to the moon and we just add up these numbers and boom we get the total potential energy it's a lot easier to deal with and that's why energy is so nice compared to forces they can be related to one another so you don't have to give up one concept in favor of the other the two can be related to each other finally let me close out this lecture video with a question but seriously what is gravity really? at this point we've been very successful in taking the law of gravity and thinking about falling near the surface of the earth and obviously we're going to look at applications of the law of gravity to planetary motion and other kinds of things like that that involve gravitating objects you might be tempted to think that wow we've got this law of gravity we know what gravity is we have totally explained gravity but we have not explained gravity we can tell what force two bodies will exert on each other through the gravitational interaction we can use equations of motion Newton's second law we can compute accelerations, displacements, times you name it but describing a thing is not the same as understanding a thing and the law of gravity gives us insight into the nature of the force but not its cause Galileo managed to show through careful experimentation that objects falling in the gravitational field of the earth do so at a constant rate of acceleration however he could not explain how this acceleration was conferred he could describe it he could quantify it he could prove it but he could not tell you why it happened Newton working out not only the laws of motion but also the law of gravity explained Galileo's observations why is it that the mass of the falling object doesn't matter it's because the law of gravity gives us the force, the acceleration comes from Newton's second law and the mass of the falling object cancels out of both sides of that equation we looked at that gravity is the mutual attraction between two masses and it can happen even over vast distances I'll have more to comment on how vast or not in a moment and no intervening material is required nothing transmits the force you don't need contact in order for this force to act, it's weird the only force at the time let's say of Newton that could be described that didn't require physical contact now magnetism was certainly well established but understanding magnetism and really understanding electricity that would have to wait many many many more years until people would work out the mathematics of that but let's go back to gravity so who was it that finally figured out the why of this force? where does this force arise from? that would have to wait until Albert Einstein Einstein spent the better part of about 10 years from roughly 1905 to 1916 grappling and I mean grappling he calculated he learned mathematics that he'd skipped out on in graduate school because he didn't think it was important at the time boy did he pay for that later one of his best friends from school he invited him to come and educate him in the geometry class material that he'd skipped out on in graduate school thinking it wasn't useful for anything oops there's a lesson in paying attention in math class he failed he made calculations using his new ideas that if they had been tested at the time he published his papers Einstein would have been the laughing stock of the scientific community because he would have gotten it wrong but because it was so hard to test his ideas it took a long time and that gave him a lot of time to work out all the mistakes all the details and it was really in 1916 that it's recognized that he published the seminal descriptive work of what we now know is his general theory of relativity really a general theory of space and time it was he who would finally shed light on the physical origin of this mutual attractive acceleration between any two bodies and more importantly he distinguished his own work from that of Newton on whose shoulders he would have been building these ideas on the shoulders of many others as well but let's pick on Newton here in particular he had insights through his understanding of gravity that were not the same numbers that Newton would have predicted from his law of gravity and Einstein turned out to be right but that's a whole other story let's focus on the essentials here in the interest of wrapping up this look at gravitation Albert Einstein was extremely famous for what we now call gedunkin experiments these are experiments done in the mind they're thought experiments you can't easily do them in the real world but you can run them in your head you can use the known observations and laws of nature and you can kind of run the experiment in your head and think about what the possible outcomes would be so let's conduct a thought experiment and it's actually similar to one that occurred to Einstein while he was employed at the swiss patent office in barren switzerland between graduate school and when he actually got offered his first faculty positions after his his big year in 1905 it's this thought experiment or a version of this experiment that inspired his original thinking about gravity and led him on this roughly ten year quest to figure out what gravity actually is here's our experiment you awaken suddenly in a room that has no windows and no doors you put your ear to a wall you hear no sounds whatsoever no vibrations are coming through the floor the walls you hear no noises it is utterly soundless and silent obviously at this moment you are extremely desperate to figure out where you are this is like the opening scene of a horror movie and to add to the horror on the floor is a single red ball desperate to figure out where you are you pick up the ball and you hold it out and you drop it and you do this a bunch of times and you know approximately what the height is typically between your hand when you out stretch it and the floor and you use the watch that you have on your wrist and you time the fall over and over and over again from the same height and you work out the acceleration of the ball and you're relieved to find out that it's 9.81 meters per second squared why are you relieved well you've narrowed your predicament so you think down to being trapped in a box somewhere on the surface of the earth after all you're observing the acceleration due to gravity that we have come to know and love 9.81 meters per second squared and so you exclaim I must be on earth this is great right you've at least narrowed it down to the one planet in the cosmos and after all there's clearly a gravitational present you were laying on the floor of this mysterious soundless windowless doorless room when you then stood up you had to work to get off the floor that felt like gravity when you drop the ball it accelerated down to the floor of this room at the right rate 9.81 meters per second squared but are you actually on earth how do you know does this one observation alone answer this question definitively Einstein's insight was to realize that it did not and this was his great insight into gravity in a nutshell this was his insight sure you could be in a windowless doorless soundproof box on the surface of the earth experiencing earth's gravitational acceleration or you might be in a box that's strapped to a rocket that is shooting you through empty space far from any planet but the rocket is accelerating you at 9.81 meters per second squared such that when you wake up on the floor of this box you try to stand up but what's happening is of course is that as you try to move away from the floor the floor is moving up toward you and you feel a force that feels just like gravity and then you pick up the ball and you drop it and it's not that the ball is being pulled to the floor by some planet's gravitational field rather it's that the ball is now free from physical contact with you or the walls of the box and it runs into the floor of this room as the floor is accelerated up toward it by the rocket engine from your perspective in this non-inertial reference frame you are absolutely unable to tell the difference between these two scenarios standing on the surface of planet Earth in this box or being in a rocket box going through empty space where the rocket is accelerating you in what you would call the upward direction at 9.81 meters per second squared causing things to seem to fall down toward the floor at that same rate Einstein's great insight was to realize that there is no observable difference between being in a gravitational field and experiencing an equivalent acceleration now this alone does not lead you to the general theory of relativity the general theory of space and time but it sets up a principle of equivalence between gravity and acceleration that allows you to have a key insight that gravity is just another acceleration and if you could just work out where it's coming from you would then know the answer to the question what is gravity it was this thought experiment one like this that set him on the path to unraveling the nature of gravity a ten year odyssey a deep dive into mathematics which at the time was considered novel but not applicable and now it is some of the most applicable mathematics is it possible for us to do things like build the GPS system and understand the collisions of black holes billions of light years away from earth let me give you just a glimpse just a tiny taste of what's going on here and I am really waving my hands at this point I want to link what's going on to things that you are already familiar with and so to do that I have to sacrifice a little bit of accuracy but in the interest of making a connection with things that are more familiar to you so here we go what is an acceleration acceleration is the second derivative with respect to time of a spatial coordinate it's this equation acceleration is d squared x dt squared it's the first derivative of velocity velocity is the first derivative of spatial coordinate acceleration is the second derivative with respect to time of spatial coordinate now in calculus you learn, usually fairly quickly that there is a geometric interpretation of the first derivative we looked at it a little bit early on in this course the geometric interpretation of the first derivative of a function with respect to some variable it is the slope of a function in this case function x at some given location in coordinate space in this case time t so if we were talking about the first derivative the picture we should have in our mind is that it's the slope of a line tangent to the function at a given point now the second derivative also has a geometric interpretation in calculus it tells us about the degree of curvature of a function at a given point it can tell us whether or not the function is curving downward upward or not curving at all so accelerations appear to be related to some kind of curvature of something so then what must gravity be if gravity is indistinguishable from an acceleration then its origin must be similar to that of any acceleration and specifically Einstein realized that it's a consequence of the fact that space and time can have curvature they are not flat grids on which the events of the universe play out passively rather they can bend and curve in response to matter and energy and their curvature influences matter and energy and the space time of our universe is as much clear in the life of the universe as the matter and energy that plays out on this great stage space and time themselves can no longer be considered as separate in fact in Einstein's description of the universe which still to this day holds they are really aspects of a singular four dimensional volume known as spacetime gravity is merely the curvature of spacetime where does that curvature come from we'll look at that in a moment but you can begin to see why it took Einstein over a decade to work out the mathematics that demonstrated this connection between space time curvature and gravity so where does the curvature come from well according to the general theory of relativity the general theory of space and time if there's matter or energy present in a volume of spacetime spacetime will curve with a degree proportional to the amount of energy and matter in that volume if spacetime curves matter for instance will move in response matter is just responding to this second derivative it's just responding to the curvature you can think of it albeit in four dimensions as rolling around on a curved surface so why does the earth attract the moon well the explanation from the general theory of relativity is depicted cartoonishly over here on the left the earth curves spacetime the moon curves spacetime too but to a lesser degree the moon goes in orbit around the earth in the same way that a marble would roll in a dimple around a bowling ball that's parked in a big rubber sheet spacetime is warped or curved by the presence of mass the earth in this case and the moon is simply responding to that curvature it's simply moving in response to that curvature there's no funny thing that's reaching out and grabbing the moon the moon is merely responding to what space and time are telling it to do because space and time themselves are curveable so to paraphrase the late physicist John Archibald Wheeler who really had a the archetypal compact description of all of this it's the following what's going on here really where gravity really comes from is that matter and energy tell space and time how to bend and space and time tell in turn matter and energy how to move it's a beautiful dance of energy, matter, space and time all woven together into a singular framework that has been extraordinarily successful not only at describing the universe we know but predicting aspects of the universe that we didn't know existed we went looking for them and they turned out to be true now there are lingering questions about gravity and as you will see from the second semester of introductory physics gravity is not the only force that acts over a distance without the need for contact there's electricity and magnetism and if you get into the third semester physics course modern physics will go beyond that you'll also learn about other forces new forces that are very distinctive from either gravity or electricity and magnetism that lurk at least at first blush inside the nucleus of the atom itself these are the so called strong and weak nuclear forces now these other non-gravitational forces electromagnetism the strong and weak nuclear forces they have all been extremely successfully explained by something known as a field theory framework the field theory framework arose from the study of electricity and magnetism you will begin to touch on it a little bit in the second semester physics course whether it's called field theory or not and the question that a lot of modern physicists have been wrestling with now for about 40 or 50 years is whether or not gravity itself could similarly be described as a field theory is there for instance a particle that transmits the gravitational attraction between two masses much the same way that a particle photons the particle of light transmits the electromagnetic force between two atoms this is an open question many attempts have been made to write down a mathematical theory that does this and all of those attempts have essentially failed to give us testable consequences that would allow us to distinguish this idea from other ideas so this is still very much an open frontier it may be that gravity does not require a field theory or it may be that we have so far lacked the imagination and the mathematical understanding to construct a successful field theory of gravity or maybe it's something else that's going on one should keep an open mind but this is one of the frontiers that gravity has given us in our field another frontier is the following seemingly simple question does gravity behave according to Einstein's general theory at all distance scales is gravity a 1 over r squared force on the scale of the atom is gravity a 1 over r squared force on the scale of a water molecule how about on the scale of a book a planet a solar system a galaxy a cluster of galaxies a super cluster of galaxies we have tried to look everywhere gravitational measurements and everywhere we've looked so far essentially behaves according to this 1 over r squared law gravity could deviate from the 1 over r squared law it may not have this behavior at all distance scales maybe it's 1 over r cubed as we go to very very very short distances but we've seen no evidence of that so far now there's a lot of reasons that people think that maybe there should be such deviations from the 1 over r squared law which is why people look for them but so far we've not seen them we're pretty confident that gravity behaves essentially according to the law of gravitation between roughly the millimeter scale and scales that are as big as roughly the visible cosmos but does gravity really work at all distance scales how about on the size of a nucleus how about on the size of an atom we simply have not been able to test that because it's too difficult to tease out the tiny effects of gravity compared to electromagnetism and the strong and weak nuclear forces once you get down to things the size of an atom and actually once you get down to things the size of a millimeter all the little electrical and magnetic interactions between atoms in materials make it very hard to actually see the gravitational attraction between two masses so this is a very challenging problem people continue to test this question and see what's possible but right now we don't know for sure that gravity works the way the law of gravity says it does or Einstein's general theory of relativity does at all possible distance scales and sort of related to this is what happens when gravitational forces get very strong are there places in the universe where gravity could be as strong as some of these other forces because there's so much matter and energy packed into a small volume this would be highly dense matter and there are places in the universe where these situations can arise so for instance the corpses of long dead exploded stars neutron stars or black holes these are highly dense forms of matter it's certainly the case that we expect that gravity and electromagnetism and the strong and weak nuclear forces are all kind of competing with one another on nearly equal footing in some of these places but of course we don't have a neutron star or a black hole that we can just go and play with and so doing experiments on this has been very difficult although those experiments thanks to recent developments in astronomy are getting more feasible and maybe a little bit easier as time goes on and maybe we'll learn something about this from these objects we also think that the universe was extremely dense and compact at the beginning of time the so-called big bang that started off the universe would have started from a state of matter and energy that was extremely compact very dense so that's a place where gravity could have been quite strong in comparison to its cousin forces and until we can understand how gravity works at that kind of density and size scale we can't really understand the beginning of the universe we can get very close to within millions of a billionth of a second or so after the universe came into being but to go back further than that we just lack the data and many physicists are working to try to figure out what data is needed in order to answer that question where would we look in the cosmos to get answers now about things that happened 13.7 billion years ago so I hope that you've enjoyed this slightly deeper dive into a singular force one of the fundamental forces of nature gravity in this section of the course we have explored a few things we have gone deeper into gravity as a force we have looked at the mathematics describing the properties of gravity as a force and we've used the law of gravity to reconnect to old ideas about constant or nearly constant acceleration due to gravity near the surface of the earth independent of the mass of the falling object and the potential energy of objects held above the surface of the earth near the surface of the earth in a gravitational field we've seen how to connect the new ideas to these old ones and show why those old ideas were true in the first place we've connected the terrestrial and the cosmic through gravity gravity acts in principle at all distance scales it's just the degree of its effects may or may not be masked by other fundamental forces also competing in the same territory and we've glimpsed the nature of gravity as the curvature of space time and we've looked at some of the modern puzzles in gravity as a fundamental player in the cosmos one of the hands that shaped the universe as we know it today this is a launching point for many other things that you can do in understanding nature understanding the law of gravity and being able to play around with potential energy and force with the law of gravity is excellent practice for playing around with other fundamental forces of nature like electromagnetism so I hope that you've taken seriously this little dive into gravity and we will use what's going forward to connect to other ideas of motion and look more deeply at different kinds of motion where gravity can play a role