Program 17 - "Gravitation"

 

MusicWhat's happening with this weight?Is it getting heavier and lighter as it moves up and down?Or is there some relationship between weight and motion?Or is there something else going on here?MusicSilico: "We are back with Science 122, The Nature of Physical Science.This is the telecourse that knows how to take your weight away.This is program 17, Lesson 3.2, Gravitation."Before we're done with this program we will have seensome of the implications and uses of Newton's law of universal gravitation.We will see how weight is explained by combiningthe second law with gravitation.We will see why astronauts are weightless, and the relationshipbetween inertial and gravitational mass, how the tides are theresult of gravitation, how atmospheric pressure is causedby the weight of air molecules, and how Halley used Newton'sequations to predict the return of the comet that bears his name.We will learn why planets are flattened at the equator,how the motions of projectiles and satellites are related,how Newton's theory of gravitation was tested and usedto find new planets, and how gravitation is usedto navigate between the planets.Last, but not least, we will learn how the gravitational equationwas used to find the mass of earth and other planets.

 

Here are the objectives for today's lesson.These objectives are also in the Study Guide at the beginning of the lesson.Before you begin to study the lesson, take a few minutesto read the objectives and the Study Guide for this lesson.Look for key words and ideas as you read.Use the Study Guide and follow it as you watch the program.Be sure to read these objectivesin the Study Guide and refer to them as you study the lesson.Focussing on the learning objectives will help youto study and understand the important concepts.Compare the objectives with the study questions for the lessonto be sure that you have the concepts under control.That was a very good introduction again my silicon friend.Now one of these days you're going to be so goodat this that I won't have to be here at all.You can just do the entire program.I can just go some place else and do something.Don't, don't say it, I know what you're going to say, just don't say it.Oh, hello.

 

Today we're taking about gravitation.And you know Newton's theory of gravitation was acceptedimmediately by the scientific community although it couldnot be quantitatively tested in the laboratory.Think about this for a minute.How do you quantitatively test gravity.I mean, Newton had done this with the moon, right?He had taken the moon, he'd looked at the tripodal accelerationand considered the moon to have a circular orbit and saw that it matched.But still, Galileo had established the precedent for experimental science.So how do you go into the laboratory and you take aweight here and you take a weight here and you testhow strongly they attract each other.Well, actually this was tried in the laboratoryin Newton's time, and it failed miserably.You simply cannot easily go into the laboratory and take twoweights and measure the force of attraction between them.Why not?But as it turns out we'll see later on that the gravitationalconstant that constant of proportion that relateseverything is such a small number.

 

The gravitational force is really, really very weak.In fact, it only turns out to be important when you haveat least one very large object, like the earth or the sun or the moon.With small objects the gravitational attraction simplyis so small that it's almost impossible to measure them.I say, almost, this has actually been done now in the laboratoryand that will be the last topic of today's program when we talkabout Cavendish weighing the earth.The compelling argument by Newton left few doubts.He had done this so well.He had sewn up all the seams with the combined inductionand deduction of the analytic geometry.The synthesis left very few holes and even the most hardenedskeptics were easily convinced that the theory of gravitation worked.First of all, it explained many phenomena such as freefalland planetary motion which had been problems from the beginning of time.It also brought under one principle other phenomena such as thetides, the flattening of the planets, and earth included, at the poles.

 

So, Newton just gave such a compelling argument,he used the mathematics, he had the analysis, everything justfell into place, and there was really no doubt.Some of the doubts did turn up later on as we'll see later onin the program when there were some things that happened thatmaybe Newton's laws were incorrect.But I'm getting ahead of things again.So, let's move on and see what happens whenwe try to explain an object's weight.The theory was not just successful in explaining themotion of the planets, it was truly universal.By that we mean that it not only applied to the earthly realm aswell as the heavenly, but it also unified all types of motion,projectile, violent and planetary while it unifiedthe solar system under one law.At the same time it provided a unified problem solving methodwhich applied to all of physics, not just astronomy and mechanics.So let's look at these things a little bit one by one.

 

The universe was formally divided by Aristotleinto the heavenly and sublunar realms.Newton actually showed here that the same laws appliedto the earthly realm as to the heavenly realm because thesame gravity worked on earth to hold the apple to the earthas it did to hold the moon in its orbit.He also unified all types of motion.Now remember that Aristotle had classified motion into four types.He classified them into projectile motion,violent motion, planetary motion, and alteration.Well, Galileo had showed that projectile motion and I shouldsay that projectile motion was the same as a combinationof violent motion and natural motion or freefall motion.Newton went one step further and showed that celestial motionor the motions of the planets also fit under the same type of description.The same equations, the same descriptions of velocityand time and distance and so forth were used in describing theplanetary motions as were used in describing motion on earth.Galileo's equations, that is.Newton's system of gravity also unified the solar system under one law.Now this is sort of the same as saying that it brought togetherthe heavenly and the sublunar realms, but it's not quite the same.The reason why it's not quite the same is that here we're lookingat a way of simplifying the calculations of the planetary orbits.

 

Now Kepler's laws had done this to a degree, but nowhere nearthe way that Newton simplicity did.Because now it's possible to look at one planet and simply applyone equation and predict where the planet would be to whateverdegree of accuracy you were capable of calculating.It simplified the calculations of the planetary orbits.It was not just a descriptiveand a predictive principle, but also a causal principle.In other words, there's now a relationship between a forceexerted by the sun and the motion of the planets.The last thing it did here under the concept of universal wasthat it unified the problem solving methods of physics.That means that the same methods of analysis that Newton hadapplied to analyzing the motions of the planets could be usedin analyzing the motion of anything, whetherit's the planets or something here on earth.So, one of the concepts we see or one of the ideas we see comingout of this is that the "uni" in the word, universal.It's uni-versal,it's uni-fied.And Newton had basically unified the concepts of astronomyand physics, both in the heavens and on earth.Now it's time to start talking about weight.What exactly is weight?

 

Well, now we can see that weight is the gravitational forcenecessary to accelerate a mass at the rate of "G."It is on one hand a consequence of the gravitational equationapplied to the earth, "M-1" in the equation and an object "M-2"in the equation, and the distance from an object like yourselfor myself or the apple at the center of the earth,I should say at the surface of the earth, and the center of the earth.In other words, the distance from the center of the earth.Remember that Newton had shown that the gravitational distancehad to be measured from the center of objects.So actually it would be from the center of you to the centerof the earth, which is roughly the same as the centerto the surface, because you aren't really very big.We can also see this as a second law relationship.

 

Now remember the second law is "F" equals "M-A."Where "F" is the gravitational force and "A" is theacceleration of an object which we have been calling "G."So, if we go to the ELMO, I think we can get an idea of how this works.Let's go to the ELMO and see.OK.Here, remember that the second law says that "F" equals "M-A."Now this is a general equation which says thatforce equals mass times acceleration.Where the force can be any force whatsoever, whether it's theforce of a spring or the force of gravity, or the forceof something else, and "M" is a mass and "A" is the acceleration of that mass.So, what we see here is that we can specify this equation thatwe're going to look at the mass of an object near the centerof the earth and we're going to look at what force is requiredto accelerate that object at the rate of "G" where "G" is theacceleration of gravity which we've accepted dueto Galileo as 10 meters per second.

 

Now, those of you who are picky out there,this is actually 9.8 meters per second squared.And if you want to get really picky, it's 9.79 and a fractionand it varies slightly from place to place on the earthdue to the differences in the earth's density.So, for our purposes we're going to take the accelerationof gravity to be 10 meters per second squared.Just to round things off, 10's a nice number.So, we're looking at any given mass here on the earth'ssurface, and we're looking at what force is required to acceleratethat mass downward at the rate of "G," 10 meters per secondsquared, and we're simply going to say that, that force is equal to its weight.

 

Now we'll see here in a little while that this weight isactually the gravitational force acting on the object which canalso be written as, according to the gravitational equation,where "G" is the gravitational constant,"M-1" is the mass of the earth, "M-2" is the objectand "R" is the distancebetween the center of the earth and the surface.So, this, again, is the general expression for gravitational force.We can also see from this description of weight whyit is that all objects fall at the same rate.Remember, Galileo had shown this to be true,that all objects fall at the same rate.Using the second law of relationship we can see exactly why that happens.

 

Now, I won't say exactly why it happens,but we see at least mathematically why it happens.So, let me show you on the ELMO.OK.So we turn this over like this.So here's the idea.If we write that the weight of something is equal to masstimes acceleration of gravity, that must mean that theacceleration of gravity is equal to the weight divided by the mass.Right?These two are equivalent statements.Even those of you who are algebraically challenged canprobably see that those are equivalent.If you have trouble with this, simply divide both sidesof this equation by "M" and you'll wind up with this one.So, here we see that the rate of gravitational accelerationis actually the ratio of the weight to the mass.So, let's take this now and look at weight and massand write this equation a little bit bigger.

 

So now, suppose I have an object which has a certain mass,and we know that it's going to fall downwardaccelerating at the rate "G."Suppose that I take now another object that has twice as much mass.Galileo had shown us that this object will also accelerateat the rate "G," in other words, 10 meters per second squared.From this relationship we can see why this happens.Here's why.On one hand the weight of this object is thegravitational force acting on it.So, "W" over "M" for that object determines its rate of freefall acceleration.On the other hand, this object has twice as muchmass, but it also has twice as much weight.Right?It has twice as much mass, which means that it resists beingmoved with twice as much inertia.It also has twice as much as gravitational weight whichmeans that there's twice the gravitational force acting on it.

 

So what happens here is that the object which is twice asmassive has twice the force pulling on it but it also hastwice the resistance to motion and so the two factors cancelout and we find that the rate of acceleration "G"is the same for both of the objects.In other words, the two objects accelerate at the same ratesimply because the gravitational force of them is doubled,but also there is this inertial resistanceto motion is doubled at the same time.So from here we see that weight is not the same thing as mass,although the two are closely related.It's hard for us to separate the two having grown up herein a gravitational environment where wehave a hard time separating them.But mass is an intrinsic property of an object.This is invariant.It's the same anywhere.It's related on one hand to the ability of an object to exertmutual gravitational attraction, and on the other hand, to resist motion.It remains constant, even during "weightlessness" in freefall or in orbit.This has been painfully recognized by the astronauts who found outvery early on that if you run into a wall, even when you'reweightless, it still takes a force to stop you, and it still hurts.

 

 

 

So now we have to ask the question, "How is it thatyou can be weightless in space?Because we know that the gravitational force extendsindefinitely because the factor one over "R" squared getssmaller and smaller, but never becomes zero.So I'm going to look at this in terms of three different thingsand we'll build our way up to the idea of freefall in satellites and astronauts.We want to look at elevators, projectiles and satellites.So, let's first go to look at elevators.On the screen is a picture of what might happenin an elevator if the cable broke.Now, how would you know if you were in an elevator and the cable broke?What would you experience inside the elevator?Well, you might look at it something like this.You see the woman on the right here experiencing whatwe might call a freefall inside the elevator, and what we see isthat she and the elevator are accelerating downward at the same rate.What this means is that basically the reaction force is missing.

 

Now, normally if you're standing in an elevator and the elevatoris at rest, the floor of the elevator exerts an upward forceon you which allows, which holds you up.Of course, if the elevator is accelerating at the samedownward rate as you are, then the elevator floor cannot exert that force.Try to imagine standing on a trap door and suddenly the trap doorfalls away from you at exactly the rate of "G."Well, then you and the trap door will be acceleratingat the same rate, so the trap door will no longer exert an upwardforce on you and you will fall at the rate of "G" and should youhappen to drop your keys or your purse as the young lady standinghere in the elevator, you would find that they don't fallto the floor, because, they, too, are acceleratingdownward at the same rate as you are.We might say that now the gravitational forceis unbalanced and so acceleration occurs.OK.So, in elevator, if the elevator suddenly, the cable suddenlybreaks, you will know because you will appear to be weightless.By the way, I should point out here that there has never been a caseof an elevator where this actually happens, because elevators haveall kinds of safety feature built into them so that if the cabledoes break, there are little hooks and things that catch you.So don't be afraid to go in an elevator just because of that.OK.

 

Now let's take a look and see how projectiles fit into this.Projectiles actually are very much like the elevator.Remember, Galileo showed us that in a projectile you'refreefalling basically while you're moving forward.So, if the elevator was not just freefalling downward,but was shot out of a cannon, say for example,it would still be in freefall wouldn't it?And it would be moving forward and the effect inside theelevator would be exactly the same as it would beif the elevator were simply falling.In other words, you inside the elevator would not be ableto tell in the absence of air friction, whether or not theelevator were simply falling downward or whether youwere moving as a projectile.This, by the way, is how they experience fake weightlessnessin such movies as Apollo 13 in what they call the "vomit comet."Which was an airplane which would simply go into an arcwhich would allow the people inside it to be in freefallfor that length of time when they weightless.Of course this doesn't last very long, because the airplane can'tstay a projectile for too many minutes, otherwise it will crash into the ground.OK.So, projectiles then are sort of like freefalling elevatorsexcept they happen to be moving forward.What about satellites?Let's see what happens with satellites.The picture on the screen now is Newton's original drawingof how to launch a satellite from a mountaintop.

 

Now, I want to point out that this was drawn by Newton himself350 years ago, a long time before anybody ever launched a real satellite.Let's go to the ELMO and I'll show what this is about.OK.So here's Newton's thinking on this.This is sort of an expansion of the flat earth idea.So, if you were to stand on top of a tower and launch a projectile,we discovered that the projectile would follow a curved pathand land some distance away from the tower.The distance away that it lands depends upon how tall the toweris, and the speed at which it is projected.So, the faster you project this, the greater distance it willtravel in the amount of time it takes it to fall that, from that height.So Newton asked, "Well, how can you make it gofurther without increasing its speed?"The answer is very simple.Suppose you don't launch it on a flat surface,but suppose you launch it on a sloping surface.In other words, the tower sits at the top of a hill and you nowlaunch it horizontally off the top of the hill.It will follow the same projectile curved path, but it will go thismuch further distance because of the fact that it falls a greaterdistance before it reaches the surface.OK?

 

So now let's extend this.This is one thing that's interesting because Galileoin all of his wisdom still considered things on a flatperspective, even though he knew that the earth was round.So, let's look at this now and see if we can reproduce Newton's drawing.So, this is what Newton was getting at in this picture.Here's a picture of the earth, a round earth.And now let's imagine that same tall tower.And let's launch the projectile first of all, this is notto scale, but let's launch it at a fairly small speed.It will fall to earth in this way.

 

So now if we launch it a little bit harder, a little bit fasterI should say, it will travel a greater distance but thedistance it travels will take into account the curvature of the earth.If the earth was flat, it would have traveled only this muchdistance, but because the earth is curved, it actually landsaround the curvature of the earth, and this far away.So what happens if you keep launching the projectile at faster and faster speed?Well if you launch it faster, and it will go this much further,if you launch it even faster, it will go this far, and finally,if you launch it at exactly the right speed, Newton reasoned,it will go around and it will come back exactly to the same spaceit was launched from because the rate at which it falls backto the earth is exactly the same rate at whichthe earth's surface curves away from it.

 

Newton even went so far as to calculate how fast thiswould have to happen based upon the curvature of the earth,and he realized that if you launch something near the earth'ssurface, again ignoring air friction, at a rate of about,in modern terms, about seven kilometers per second,that its speed would be just such that it would fall backto the earth at exactly the same rate the earth curves awayand it would then describe a circular orbit around the earth.You see what he's doing here.He's basically taking the concept of the moon as a satelliteand saying that under certain conditions,a projectile can become a satellite.And we're finding that the projectile motion or satellitemotion are actually one and the same.You can also see that the satellite or this projectile as it falls isstill pulled back to the earth by the acceleration of gravityas in triple acceleration, exactly in the same way that theprojectile as it falls from the tower is pulled back to earth in Galileo's system.

 

I want to come back to an interesting point now thatI brought up in an earlier section.That is the idea of this equivalence of mass.Einstein recognized that this equivalence of two differentproperties of mass was a significant principle.Here's what it means.It seems that when we talk about mass we're talkingabout two different things.On one hand there's mass as Newton described it in thesecond law, that is, mass as a measure of inertia.Mass is related to the amount of force in acceleration.A certain amount of force causes a certain massto accelerate at a certain rate.On the other hand, mass appears to be related to gravitation.That a certain mass will provide a certain gravitational attractonin conjunction with this mutual gravitationalforce that Newton described in the equation.The question is, "Why are these things equivalent?"

 

Now the fact that all objects fall at exactly the same rate asI showed you in the earlier section, means that the two things must be the same.Right?Because you double the gravitational force on a mass,you also double its inertia, the two things cancel outand, therefore, all things fall at the same rate.Einstein saw this to be a very important principle and you canbet that in the 350 years since Newton, it's been tested overand over and over and over again to see if all thingsreally do fall at exactly the same rate.The question I want to leave you with, by the way before I do that.Einstein went on to show that if this equivalence principle istrue, then it defines a whole new way of looking at the conceptof gravity, which he went on to describe as general relativity,which we won't have time to consider in this programor in any of the programs actually, but it's a veryfascinating topic if you want to go and look it up.

 

So, the question I want to leave you with, sort of a philosophicalquestion on this is, "Why are these two kinds of masses the same?What does it mean that gravitational mass, that is,the ability to attract something is equivalent to inertial mass,that is, the ability to resist being moved?"This is one of those philosophical questions that we don't wantto get hung up in because we don't want it to hold usback from understanding other things.But I think it's a very interesting one.One of the phenomena that Newton explained after he had put forthhis theory of gravitation was the tides.It had been known since ancient times that the moon affected the tides.In fact, it was known that the full moon and the high tides,not exactly correspond, but occur within a few minutes of each other.I should say occurs within a few minutes of the timethat the moon passes directly overhead.Using the gravitational equation we can easilysee where the tidal force comes from.It comes from the fact that one side of the earth is much closerto the moon than the other side of the earth.In fact, you may remember that the gravitational force is an inverse square force.So, that means that this side of the earth is about 4000 milescloser to the moon than this side.

 

Now, that's a significant amount because the 4000 miles is squared, right?The difference between the distances is squared.So, what actually happens here is that the water on this sideof the earth is more attracted to the moon than the earth itself is.So the moon pulls the water away from the earthcausing a tidal bulge on one side of the earth.At the same time the water on the opposite of the earth isattracted to the moon less than the earth itself is.So on that side of the moon the earth is pulled awayfrom the water in the same way that it's pulled, the water'spulled, toward the moon on the other side leavingan equivalent bulge on the other side.You might want to consider the possibility of what mighthappen if you tied a string to a flexible rubber ball and spinthe string really fast and spin the rubber ball really fast around your head.The ball is going to squash and distort.That's exactly what happens with the earth.

 

Now I want to point out that the solid earth itself alsoundergoes some tidal movement, but it's much, much smallerthan the tidal movement of the waters.Another thing now is atmospheric pressure.Newton was the first person to explain whereatmospheric pressure comes from.Galileo and his students had measured atmosphericpressure and recognized that there is air.Which, by the way, is a major accomplishment.It's one of those things again like trying to geta fish to understand about water.The fact that we are surrounded by air pressure.But Newton went on to explain it simply that air pressure is theresult of the weight of the atmosphere above a given point.That the air molecules themselves are held to the earth by gravity.So if an air molecule is trying to escape into space, like anyother projectile, it's simply pulled back down to earthand must be moving at a high enough rate of speed to leavethe earth's atmosphere if it's going to escape.So, you also note that the air pressure decreases with elevation.

 

Edmund Halley who was a friend of Newton's used Newton'sequations also to figure out the appearance of the comet.The comet had been last observed in 1607.In fact, it was one of those that Kepler had observedwhen he was working with Brahe's data.You may remember that there were three cometsthat appeared within a 20 year period there.How they did, basically, was to use Newton's equationsto predict the comet's orbit based upon observations which weretaken in 1607 using Brahe's instruments.Now, interestingly enough, Newton had made a special pointin the "Principia" of giving a very precise geometric methodby which the orbit of a comet could be ascertainedor predicted by making three simple observations.Now this is an amazing thing.When you look at this in Newton's book it's just incredible thatsomeone could even come up with this.Remember, of course, that Newton had publishedthe book at Halley's insistence in the first place.So, Newton goes back and gives this methodwhereby Halley could predict the comet coming back.I should also point out that Halley made this prediction of whenthe comet would appear which is why it bears his name,and he died before the comet ever actually came back.It has appeared at a fairly regular rate every 76 years since then,and, of course, is named now in Halley's honor.You may remember that Galileo had noticed first of all thatthe planet Saturn was not perfectly circular.That it was flat or squashed, and this was one of the things thatconvinced him that the idea of spherical perfectionin the heavens could not be true.

 

By Newton's time it was known that allof the planets bulge at the equator including the earth.Now the difference between the earth's equatorialcircumference and its polar circumference is only a few miles.So for all practical purposes, the earth is still spherical.But, it's enough of a difference, that we have to account for it.Newton in his genius was able to recognize that again,by combining the concept of gravitation and the conceptof the second laws and the laws of motion, that he could explainvery easily how the planetary bulges come about.Let's go to the ELMO and I'll show you how he did that.Here's what happens.Suppose you start out with a circular planet.I know that, that circle would not satisfy Plato as far as theperfection goes, but let's say that it's a circularplanet, and I'll sort of dot in an equator here.So the planet is rotating on its axis which it has to do,of course, in order to, the earth, but all the planets too, in orderto make it around once every 24 hours.Of course, the rotating planets are a part of the concept of heliocentrism.

 

So, if the earth is rotating, what does it take to keep the earth in its orbit?I should say to keep the pieces of the earth from flying off into space.Remember that Aristotle had claimed that if the earth wasrotating, then things would fly off into space likewater being flung out of a wet rag.So, what actually happens here?Well, I think you can see that if the planet is rotating,let me actually go to a another picture here.I should say make a little space.If the planet is rotating, looking down at the top of it now,like this, then in order to keep the pieces of the earth from flyingoff into space, there must be some force which pulls them towards the center.What force is that?Well, it must be the force of gravity, right?What holds the earth together in the first place?Well, it's the fact that all of the pieces of the earth aregravitationally attracted to each other and so it sort of causesthis lump of rocks, or this clumping of rocks in space.So, as the earth is turning, the part near the equator is turning fastest.I think you can see that, can't you?From the top picture, here's a longitude line on the earth,and here's a series of longitude lines like the sections of an orange.So, a point here on the equator has to travel this much distancein a certain amount of time, whereas, a point near the polesonly has to travel this much distance.So, a point near the equator is traveling at a muchfaster rate than a point in the poles.I think you can see it on here as well.That in a certain amount of time a point here on the equator hasto travel this far, and a point near the pole has to travel this far.So, a point on the equator is traveling much faster.

 

Newton recognized that the acceleration required to keepsomething in a circular orbit is proportional to the square of the velocity.This is the formula for centripetal acceleration.So, if the velocity of a point here is twice as fast as it is here,that means that the acceleration requiredto keep it in a circular path is four times as great.So what does that mean?Well, let's see what it means.It means that here at a point traveling here near the equatorneeds a greater centripetal acceleration to keep itin place than a point near the pole does.So, what happens?Well, when the planet spins, it, I'm going to exaggeratethis, it becomes elongated.So, here's again, an equatorial view of the planet.Here's the North Pole and here's the South Pole, so that as theplanet's spinning, it takes much more force to hold this partin orbit, than it does to hold this part in orbit.So, I guess in essence what we're saying here is that Aristotlewas essentially correct that these parts of the planet near theequator are flung off into space so this causesa bulge or a flattening of the planet.

 

I should also point out that if you're standing hereat the equator as opposed to someone standing hereat the poles, your weight is slightly lessat the equator than it is at the poles for two reasons.One is that at the pole you're a little bit closer to centerof the earth than you are at the equator.And secondly, the spinning of the earth on its axis does tendto throw you off into space which reduces your weight, although not by much.The difference here between your weight at the equator and the pole.If you weigh 200 pounds, the difference is weight is a few ounces.But it's detectable.OK.

 

Now, you may remember that Newton has already shown usthat projectile motion and satellite motion arebasically the same kind of motion.But we can go further with this.Now, in this case I want to take you a little bit into the futurewhich is more like now than in Newton's time to see how it isthat we can measure or, I should say use, Newton'sequations to talk about satellites.First of all I want to point out that much of the Cold War,some of you older folks may remember the Cold War,was about intercontinental ballistic missiles aimedat various cities from various locations.The trajectories of those missiles were calculated exactlyby using Newton's equations, much in the same way that Galileo'sprojectiles could be used to predict where things were going to land.The difference is, of course, that you had to take into accountthat as the missiles rise higher and higherinto the air, the force on them is diminished by the inverse square.So you have to that, the changing gravitational acceleration into account.On the other hand, on a little more upbeat note, satellites whichare launched even today are still launched according to Newton's equations.

 

Let's go to the ELMO and I'll show you how this works.From Newton's equations, let me draw a picture of the earthin here like this, from Newton's equations we know exactlywhat speed it would take to launch a satellite from a certain height.Remember the picture I showed before of Newton's mountainwith the satellite moving around it from that height.So, basically, we launch a satellite today exactly the same way.The difference is that we don't launch it horizontallyfrom a tower or a mountain, we launch it from another rocket.So, what we do to launch a satellite is to send the satelliteup to a certain height with a rocket and the rocket,of course, it requires a lot of energy to do this,but once the rocket gets to a certain height,we basically throw the satellite out of it.And we throw the satellite out at a certain speed in a certain direction.Now, we're not going to go into the mathematical dynamicsof this, but I think you can see here that just like Newton'smountain, once you get the satellite up, the, if you throw itout at just the right speed, it will go into a circular orbit.

 

Now we've already seen from Kepler's laws that a circularorbit is a rather special case of a conic section.Remember that the conic sections include not only the circlein the ellipse, but also the parabola and the hyperbola.So, we can look at the dynamics of an orbital motion to understandor to get some sort of understanding about howthese different orbits come about.For example, there's for a given height above the earth'ssurface, there is one speed and only one speed which willallow the satellite to go into a circular orbit.Furthermore, that speed has to be attained in a perpendicular direction.In other words, it has to be exactly tangent to the earth's curvature.What happens if you launch the satellite a little bit too slow?Well, if you launch a little bit too slow, it's going to deviatefrom the circular path and it will start to fall back closerto the earth and, of course, if it's launched too slowly,it will eventually spiral back and crash into the earth some place.We prefer that, that doesn't happen when we launch satellites.But if we want it to go into an elliptical orbit, we can exactlymatch some orbital speeds so that it will swing a little bit tooclose and, of course, as it swings close to the earth it gains speed.

 

We know from Kepler's laws that when a planet is closer to the sun it gains speed.It's no different for a satellite near the earth.Or for that matter, it's no different for a rock.Think about this.If you throw a rock upwards, it loses speed as it goes awayfrom the earth and it gains speed as it comes back closer.So, the satellite will gain speed as it loops around closeto the earth and that speed is enough to throw it further awayfrom the earth and it will be in a circular orbit where it losesspeed, loses speed, loses speed, loses speed, and eventuallycomes back and if everything is donecorrectly, it will launch into an elliptical orbit.In this case I've drawn an ellipse with a semi major axis in this direction.Let me flip the paper over.What happens now.Here's the same, oh, that's not a very good circle.

 

Plato would really hate that one.Launch the rocket.Of course, we have to understand here that as the rocket itself islaunched, it will fall back to earth eventually,but we launched the satellite from here.I think you can see from here if we launched the satellitein a different direction, like this, for example, insteadof launching it exactly perpendicular to the earth's radius.If we launched it here, it's going to follow a path that's goingto take it sort of out of earth like this and it will eventually fallback picking up speed as it gets closer, gaining speed as it gets further away.And once again, it will go into an elliptical orbit.In other words, there are many, many ellipticalorbits, but only one circular orbit.What happens if you throw the object at a very high rate of speed?Let's take a smaller earth, take a little bit different, longer view of the earth.What happens if you get the satellite up here and you throwit off in some direction, like this, in a very, very high rate of speed?What's going to happen to it?Well, there's some rate of speed at which it will be moving fastenough that it will simply disappear and escape the earth's gravity altogether.

 

You might notice that if we look at half of this orbit, that thematching half on the other side is a parabola.So, in fact, if you launch a satellite at just the rightspeed, it will just escape the earth's gravity in such a waythat at an infinite distance from the earth, whoops, we use thatword, infinity, but at a very great distance away from the earth,it will simply lose all of its speed and fall backto the earth and another parabolic orbit.In other words, it will oscillate around the earth like this.Now some comets are in parabolic orbits.Those comets are not in oscillating parabolic orbits,but they are comets that fall out of orbit, usually perturbedby the gravitational attraction of oneof the larger planets like Jupiter or Saturn.So, a comet in the parabolic orbit will come near the earth,it will move faster and faster as it gets near and it will reacha close point near the sun called the perihelion of the cometor the closest point and it will disappear outinto space never to be seen again.

 

Now I want you to be aware here that when we look at the conicsections we understood that the difference betweena parabola and an ellipse is a very small one.In fact, a parabola is simply a conic section which cutsthrough the cone parallel to the side of the cone.So, the distinction between a very, very long ellipseand a very, very short, and a very, very open parabola,is not much of a distinction at all.So, when we look at planetary orbits we seethat all the conic sections are possible.In other words, we can have a circular orbit,we can have ellipses of various shapes,we can have hyperbolas, parabolas.In fact, if you launch the satellite at a speedfaster than the parabolic orbit speed, it will forma hyperbolic orbit which is a more open orbit.So, this parabola or the parabolic path is,occurs at a speed that we call the "escape velocity."OK, that's the minimum speed that's necessaryto cause something launched into space to just escapethe gravitational pull of the planet or the sun.You may recall from an earlier lesson that Francesco Sizi,the astronomer from Florence, criticized Galileo's descriptionof the new moon's Jupiter on the basis of the fact that therewere only seven planets and that if a new planet was discovered,it would destroy this entire scheme of things.Well, something happened in the late 1700s that really putNewton's theory to the test, the theory of gravity.This is what happened.Once Newton's laws were in place, and mathematicians jumpedon the opportunity to use these to calculate the positionsof the planets, it was observed very quickly that someof the planets were not where they were supposed to be.Now this sounds like the old problems coming up all over again.You may remember that all the way from Ptolemy's systemon, the planets were not exactly where they were supposed to be,and Newton's laws apparently had solved this.

 

Well, here's the problem.Newton, himself, had made more accurate and more powerfultelescopes and it was becoming easierand easier to find thelocations of the planetsdown to very, very precise locations,much more than the four seconds of arc that waspossible with Brahe's telescope.So, here's what happened.The outer planets, I should say, some of the planets wereobserved not to be in the right place at the right time.So, here's the question.Here's the quandary.Do you assume at this point that Newton's laws are incorrect?Do you assume that Newton's laws areimperfect, or do you assume something else?Well, there are actually two possibilities here.And this really put Newton's theories to the test.In fact, it was one of the best early tests as to the correctness of the theory.

 

Here's what happened.If the planets are not where Newton's theory predicts,the old way to do this would be to say, "OK, we've got to have a new theory."This is after all what had happened with the Ptolemaicsystem and the Copernican system and with Brahe's system.Right?OK.So, the other possibility is, and this is ingenious, that there'sanother planet out there someplace that's exertinga gravitational influence on the existing planets.So, at this time, the scientific community split itself into two schools.One school argued that Newton's theories had to be displacedand had to be reformulated because they obviously were incorrect.But another school took a slightly different approach.And this is where we bring in the idea of the paradigm.Remember the paradigm is supposed to guide yourexpectations and to determine the kinds of questions that you ask.So, what happened was, someone said, "Suppose Newton's lawsare correct, and if Newton's laws are correct, then there mightbe another planet out there that's causing the orbitsof the existing planets to be perturbed somewhat.

 

 

 

Let's calculate how big this planet would have to be and whereit will have to be located in order to cause exactly this amount of perturbation."You understand what I'm saying here, right?You assume that Newton's laws are correct.So then you say, "Let's figure out how big a planet would haveto be, where it would have to be to cause the deviation that's observed."Well here's what happened.Two different people did this.Fairly elaborate calculations were involved here, but certainlywell within the mathematics of the time.Within two days, count them, two days after the resultsof this calculation were published, two different groupsof observers turned their telescopes to that pointin the sky where the calculations had predicted the planet ought to be.And guess what they found?They found a new planet.The planet Uranus.

 

The next planet out from Saturn was located simply by turningthe telescopes to where the predictions basedupon Newton's laws said it ought to be.Not only that, but the same thing happened with the orbit of Uranus.It was found to be slightly off where it ought to be.So, different people collected data, did another predictionbased upon Newton's laws, and made a predictionof the existence of yet another planet.And when they did this they pointed a telescopeat the sky and guess what they found?They found another planet.The planet, Neptune, the great blue giant.This happened actually one more time.The problem with the last time, the discovery of the planetPluto, was that Pluto is really, really small.In fact, the planet Pluto is about the size of our own moon,and it's somewhere near a billion miles away.So imagine trying to locate something the sizeof our moon a billion miles away.That, by the way, is 4000 times further away than our moon.So, the planet's going to be 4000 times smaller than the full moon.Well, here's what happened.They pointed the telescopes where Pluto was supposedto be and nothing happened.Nobody saw anything.So, here the theory was put to the one final test.

 

The problem, again, was that Pluto was simplyso small and according to Kepler's laws so faraway from the sun that it moves very slowly.That means that its motion from one day to the next is almost undetectable.You know how they discovered Pluto actually?This was in the 1930s that this happened.You know what flip pages are like?You know the pages that you flip and you can make animations?This is basically what they did with Pluto.They turned their telescopes to the region of the sky wherePluto was supposed to be and they looked then at a seriesof photographs made over a period of years and basically did flippictures with them, and what they saw was onelittle dot that moved on the flip pictures.And, of course, any dot that moves relative to the fixed starbackground is by definition a planet.And it turns out now that the location and the mass of Plutoturns out to be exactly, well, within Newton's limitsof perfection, exactly what it was predicted that, that planet ought to be.This is one of the best tests and it allowed the discovery of three more planets.Suppose that you wanted to send a rocketship to a distant planet.Exactly where would you aim it to get it there?Well, we can solve this problem, too, using Newton's laws,although it does take a high speed computer to do the calculations for us.

 

I want to point out to you that sending the astronautsto the moon back in the 1970s used a computer that had muchless power than my desktop friend here.But that's another story.So, let's go to the ELMO to see how this works.Suppose you wanted to send a rocket from the earthin an orbit like this to Jupiter in an orbit like this.

 

Now we understand that if you simply aim the rocketat Jupiter like this, Jupiter won't be there by the time the rocketgets there because Jupiter's moving in its orbit.So the other thing is if you launch a satellite from the earth itdoesn't just go in a straight line, it goes in a spiral, like this,because the speed of the rocket can't be such thatyou can travel a straight line.Remember?You have to hit at least the escape velocity to get a parabolicshape and we don't want a parabolic shape, we wantsomething that's still with the sun.So, when you do this sort of calculation, first of all,you have to use Newton's laws to figure out where Jupiter'sgoing to be at the time that the satellite gets there, and whatnormally happens is something like this.That the, you launch when Jupiter's here, but you aimfor Jupiter being back here, and in the meantime earth may havegone through several revolutions, and you plan the orbit so that itleaves earth, it spirals around like this, and does a couple of orbitsaround the solar system until it contacts Jupiter over here.That's one of the problems of celestial navigation.

 

The other problem is how do you know where the spacecraft isat any given time in interstellar space.Looks like I need a clean piece of paper here.Red kind of bleeds through a little bit.So how do you figure out where the thing is?I mean, there's no sign posts and there's way, easy way to measure the distance.It's very simple.What you do is you use the rocket, the spacecraft, and you keeptrack of where it is simply by knowing where it's been.Well, here's how you do that.At any given time where it goes is dependent upon the forces acting on it.Well, what are the forces?The forces in interstellar space are, I should say, interplanetaryspace, are the forces of gravity from all the planets.So, for example, the sun might pull on itwith a certain force in this direction.And earth might pull on it with a certain force in this direction.And, Mars might pull on it in a different direction, and Jupiterpulls on it in a different direction, and Saturn, and so forth.So, you can use the gravitational equation knowing how muchdistance there is between the rocket and each of the planetsto figure out the direction and magnitude of the force exerted by each planet.You then use Newton's sense of vectors, remember vectors,how you can sum up all the different forces?And you calculate one net force acting on the rocket and you,knowing the mass of the rocket, you know what its accelerationis and what direction it's going to move and so you figure that it'sgoing to move that much in that amount of time and then amillisecond later you do the same thing over again.

 

The computer, remember, can handle all these complex calculations.So, basically we direct the motion of something in interplanetaryspace simply by looking at the vector sum of all thegravitational forces acting on it and it will move then into somenew location as a result of that and you do the whole thingover again in an intuitive sort of way, keeping track,step by step millisecond by millisecond exactly where the object is.We've kind of lost our time frame as far as keeping thingsin sequential order here, but that's OK.

 

We'll come back to that a little bit later on in other programs.The headline in the "New York Times" after Cavenndishperformed the experiment that I'm about to tell youabout was what you saw on the screen.The headline in the "New York Times," now, front pagein 120 point type was: "Cavendish Weighs the Earth."How would you go about weighing the earth?What, you take it home, put in on the bathroom scale?Do you hang it from a scale in a laboratory?No, of course, you can't do that.What Cavendish was able to do was to measure thevalue of the gravitational constant "Big G."You remember "Big G" from the gravitational constant.

 

Now, I pointed out earlier that Newton's theory of gravitywas not tested in a laboratory in his time simply because thegravitational force was too small between objects in the laboratory.By the time of Cavendish, this is in the early 1800s, a hundredyears after Newton, by the way, the, it was possible to do this measurement.What Cavendish did, if we take a look on the screenwhat Cavendish did was to set up an experimentwhereby he had two very large masses fixedin place, "M-1" and "M-2", or "Big M-1"and "Big M-2" and he suspended from a verythin wire another rod which contained two masses,"Little M-1" and "Little M-2."Now, according to Newton's concept of gravitation,the masses should attract each other in such a waythat they would twist the rod.The problem was that the gravitational attraction was sosmall between the big masses and the little masses that thetwisting of the rod could not be measured, well I should say,that the twisting of the rod was interfered withby small air currents in the room.So, Cavendish took this apparatus and sealed itinside a hermetically sealed container which he couldkeep air movements out of.He still found that small conduction currentsfrom differences in temperature between the outside airand inside the box were enough to deflect the thing.

 

So, finally, he put the thing in a temperature-controlledenvironment, hermetically sealed so there were no air currentsor anything else and was actually able to measure the deflection.He did this by a very ingenious sort of method.What you see on the picture here is, a what Cavendishwound up calling a "light scale."What he did here was to hang a mirror on this very thin wireso that as the wire twisted the mirror turned from side to side.He then shown a light onto the mirror, like this, and as itreflected off, it reflects off at the same angle that it struckthe mirror and so it bounces back and was recorded on a sticksome distance away from the mirror.Remember from our geometry, the concept of parallax, right?So notice here that the further away you get from the mirror,the larger deflection there is for a particular angle.So, he was able to measure a very small angle.The question is, "How do you turn that small angle into a force?"Well, that has to do with Robert Hooke.You remember Robert Hooke, don't you?Hooke had determined that the amount of force requiredto twist the wires proportional to the amount of twist.In other words, if you doubled the force,you doubled the amount of twist of the wire.

 

Cavendish had taken this thin wire before that, before he set it upthis way, and measured the amount of force that it takesto twist it, and then simply plotted a graph of force versusdisplacement or force versus twist, from which he could thenmeasure, I should say, calculate the amount of force for a givenangle of deflection, even if the angle of deflection was very small.And, by the way, I should point out that, that angleof deflection was on the order of micro-radiance.Oophs, I used that radian word.Maybe I should saymicro-degrees.In other words, it was on the order of thousandths of degrees.Much less precision than, much better precisionthan was possible anytime up until this point.So, Cavendish was able to measure this gravitational constant.He could measure the constant and oncehe measured that, he could figure out lots of other things.If we go to the ELMO we can show you some of the things thatCavendish was able to do with this.What happened here was that in the gravitational equationwhich says "F" is "G" times"M-1" times "M-2" times, divided by "R" squared.In Newton's original gravitational equation, remember, that themass of the earth, the mass of the moon, and the forcebetween them was unknown.In Cavendish's experiment he controlled the experimentin such a way that he knew the mass in one, he knew the massin two, he put them a fixed distance apart, so the only,and then he measured the force between them, so the onlyunknown remaining in the equation is "Big G," the gravitational constant.So, by being able to determine the values of the massesand the radius and the force between them, the onlyunknown left in the equation is the gravitational constant.

 

Now, remember, Newton hadclaimed that this is universal,meaning that it applies not just to massesin the laboratory, but also to things elsewhere.So, what does that do for us?Well, let's see.There's a couple of things that can happen.On one hand let's look at it this way.We know that the force of attraction on any object can bethought of it two different ways.On one hand, we can think of it as a gravitational equation.So, let's consider, for example, the earthand the apple sitting on the surface.OK.We know that the apple has a weight equal to "M-G,"in other words its mass times the acceleration of gravity.But, we also know that the gravitational force on itcan be calculated using the equation.So, let's see how this works.So, if I rewrite the gravitational equation this way.If we look at the mass of the earth and the mass of the appleand the square of the distance between them, which, of course,is the radius of the earth, then I know that that'sequal to the weight of the apple.In other words, the gravitational force exerted between the earthand the apple is equal to the weight of the apple.

 

Now, the mass here.We've determined before that the inertial massand gravitational mass are equivalent.So this is inertial mass of the apple, this isgravitational mass of the apple.So we can cancel that out at both sides of the equation.We're now working with this much of the equation, so what's leftnow as unknowns in this equation?Now, again, you don't have to actually do this equation,but look at how the mathematics works, and look at whatdetermining the gravitational constant allows us to do.What are the unknowns in this equation?Well, let's check this out.eThe unknowns are, let's see, let's do the variables one at a time."G" is the gravitational acceleration.That's known.The radius of the earth is known.The gravitational constant had been worked out by Cavendish.The masses had canceled out.So, what is the only unknown left?The only unknown left in the equation is the mass of the earth.What Cavendish had done here had allowed usto measure the mass of the earth.

 

I want to point out here without going into more detail,that once you apply this equation, you can also use the sameequation to figure out the mass of any planet which has a satellite.So, that means we can use the same technique to figure outthe mass of the sun, the moon and all of the other planets.In fact, we can use this exact same technique to figure outthe mass of a double star in a different galaxy, as long asthey're in orbit around each other.Cavendish's experiment in measuring the massof the earth gave us much more information.Why do we care about this?I mean who cares what the mass of the earth is?Well, for one thing, it allows us to calculate the densityof the earth, because we know what the volume of the earthis, we know what the mass of it is, and once we know thedensity of the earth, we can compare the densityof the earth with the density of other planets.So, we can ask questions like how does the mooncompare in density to the earth?How does the earth compare in density to the sun?We get an idea about which things are made outof which different materials.In other words, we gain a variety of information about the solarsystem and about the universe simply by thisone experiment done in the laboratory.

 

In this program we've seen how Newton's law of universalgravitation was tested, applied to understand various aspectsof the physical sciences, and also used to discovernew facts about the universe.There's much more, and we'll see it unfold in future programs aswe trace the nature of physical science.That's it for Program 17.Remember, when it comes to science, get physical.Slow down.You understand the effects of gravity better now?Silico: "Yes, I understand that if you want to lose weight,you can go to the moon or into orbit, but to losemass is a different story all together."Music