9 - 3 - The Famous Equation (18-29, High-Def) (1)

8/10/2019 9 - 3 - The Famous Equation (18-29, High-Def) (1)

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So now we come to The Famous Equation.Of course we're referring to E equals mcsquared.you may remember that Einstein's paper onthe Special Theory of Relativity in 1905was published in June.A few months after that, he published ashort note in September essentiallysaying.You remember that paper I published a fewmonths ago, well I've been thinking moreabout some of the things contained in it.Some of the implications and discovered avery interesting relationship.And that relationship essentially was Eequals mc squared.Now unfortunately to, to reallyunderstand this, to derive it in a, in asense, even sort in a hand waving since.Would require us to spend a couple weekson concepts of energy and momentum and afew things like that.So, we don't have time to do that.But want to just review a few key things

about it and bring a few of the, the keyconcepts as, as well.And we're going to start with the conceptof Kinetic energy.All the way back in the 1700s, early1700s scientists, they weren't reallyscientists at that time, they were callednatural philosophers.Were investigating things with howobjects move and collisions and thingslike that.And identified, had identified a certainquantity that seemed to be very

important.And over time in the 1700's and to the1800's, this quantity came to be known askinetic energy and had the form of onehalf mv squared.Where m was the mass, so you can imaginea particle, a ball, a tennis ballsomething like that mass.And then v is velocity.Okay, so we're not talking relativityhere, we're just talking everyday type ofvelocities.And also along the way as this developed,

especially in the middle of the 1800s,mid 19th century.the idea of conversation of energy cameabout.That there were different forms of energyand you could transform one form ofenergy into another form of of energy.And so they developed those concepts alittle bit more as, as well.And one key thing about this was that


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they really focused on things likekinetic energy.And what happened when Einstein camealong in 1905 and, and came up with thisidea that energy, okay, of course here'sthe famous equation, E equals mc squared.So we see, just compared to this equationfor kinetic energy masses involved andthen there is an velocity squaredinvolved in each case.This is just the regular velocity of anobject.This is obviously the velocity of light.So there is some similarity between themthere.But really what Einstein discovered was,not quite this equation, this is aspecial form of the equation that hediscovered.What he discovered was, and we'regoing to come back to see how it relatesto Conservation of Energy here in aminute.He discovered this, that energy is gamma

mc squared.Okay, gamma mc squared, in other wordshere is our familiar Lorentz factorcoming into play here.And if we play around with this a minute.Let's just remember, back when we weretalking about the Michelson-Morleyexperiment, we used something called thebinomial expansion.So we're going to bring it out of ourtoolbox one more time here.And remember it's this, if you havesomething 1 plus x to the n, some

quantity x to the n power.If x here, is much less than one.Then we can write this as 1 plus nx,approximately with that.And so we're going to exploit that herebecause, of course, gamma, so what wehave here is gamma, in our 1 form.1 square of 1 minus v squared over csquared there.but let's write that in a slightlydifferent format, as we've done before.We haven't done this recently, but beforewe could write it like this.

We have 1 minus v squared over c squaredto the minus one half power, so that'sjust gamma.So if we've got the mc squared in here,so we'll put that on the top.Okay, mc squared over that.So that's this times mc squared.So again, this is just gamma, right here,times mc squared.And, and note that this has the form


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because, especially if v is much lessthan c.So if our velocity is much less than thespeed of light.And what do we mean by much less?Well, even up to speeds like one tenththe speed of light.v squared over c squared, it's stillgoing to be a very small, very smallvalue.So we'll be able to expand this out and,and put it in this form here using ourbinomial expansion.So if we do that, what this happens hereis we've got, okay, here's the exponent,the minus one half is equivalent to the nthere.And I've got a minus here instead of aplus, but it works the same way.So, essentially, I'm going to have forthis part right here.This becomes 1 minus, and then the minusone half, and then, v squared over csquared, and, times mc squared.

Okay, now remember this is anapproximation for when v, is much lessthan c.So up until about, up until about, onetenth the speed of light, someplace inthat region.So still pretty fast, compared to ournormal everyday experience.And, so now we got a minus times anegative one half, so that becomes a plusone half.And so we'll write this out one more timehere.

So we've got 1 plus, one half v squaredover c squared times mc squared.Now we're going to bring it back overhere in a minute here just to.Looking a little bit better, okay, solet's, this is how we got there.Let's erase this part now, and say, okay.So we've shown that gamm, gamma mcsquared can be written as.We'll rewrite this here, 1 plus one halfv squared over c squared times mcsquared.If, we'll just say if v is less than you

know, about 0.1c, something like that,for low velocities in other words.But let's look at this a minute, look atwhat we've got here.1 times mc squared, that's just equals mcsquared.And I've got one half v squared over csquared timed mc squared.Well the c squares here cancel and I'mleft with plus one half mv squared.


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That is Kinetic energy.So that falls right out of this generalformula for in the low velocity limit.Well what is this telling us.Back before Einstein came along, peoplewould look at kinetic energy and otherforms of energy and would talk about theconservation of energy.What Einstein is saying with thisformula, is that there's another form ofenergy right here too.It's not only the kinetic energy that'simportant, but this mc squared factor aswell.And note that if we just had v equals 0here, then energy just as mc squared.We get our famous form of the equation mcsquared here.But what this is saying, going back toconservation of energy, is that beforeEinstein came along with this equation.Then the idea was, well, sure, you have aconservation of kinetic energy.It could go into other forms of energy,

as well.But no one really payed attention to, a,the masses involved.And what Einstein is saying, you knowwhat, you got the masses involved maybein something, you got some energyinvolved.You can actually convert in between them,and still have the whole thing beconserved.So that's what the implication of thisequation is.That, in principle, you can take some of

the mass energy here, in a sense,sometimes called the rest mass, or restenergy.Because if v is zero, you're just leftwith the mc squared part here.You can, you could potentially convertsome of this into kinetic energy.Or you could go the other way as well.As long you, if you convert some of thisit converts to this, if you have some ofthat it can turn to that.As long as energy itself, the total, isconserved, does not, does not change.

And again we don't have timeunfortunately to go into to all thedetails of this.But essentially, what you get out ofthis, what it leads to eventually arethings like nuclear fission and nuclearfusion, fission and fusion.So the idea here, [SOUND], Fission andFusion.The idea is you can turn some just


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ordinary mass, it's like it has energylocked up inside of it, it's really aform of energy.If somehow you can liberate that energy,you can turn that mass directly intothings like kinetic energy.And, turns out in atomic realm, reallythe, the nucleus of atoms.Where the protons and neutrons are.That, if you turn, say if you have your,certain types of uranium.The uranium nucleus can split apart.And it turns out that if you add up thecomponents that are left with say, tomake it simple say, it splits into twopieces versus the original uranium atom.The new atoms, the smaller constituentsthere have less mass than the uraniumatom did.And therefore, where did, where does thatmissing mass go, it turns into energy.And that's really the idea of nuclearfission, that you can split apart certainatoms, certain nuclei.

And that releases the mass energy in asense, that's stored inside there.Nuclear fusion goes the other way.It turns out that for lighter elements,say hydrogen, the lightest element.If you fuse two hydrogen nuclei togetherto create a helium nucleus, a few otherthings involved here as well.But if you create a helium nucleus out ofit, then and you look at it, the heliumnucleus actually has less mass than thetwo hydrogen.A nuclei that you use to, to put together

with.So again, where did the missing mass go?It turns into energy.And that's the whole idea of nuclearfusion.You're fusing nuclei together in a sense.And this is how the sun works, forexample.So, we're all here because of the sun, ina sense.And so we're here because of, of nuclearfusion.Nuclear fission, was developed the idea

of it, the concept of it, over a numberof years, 1920s, 1930s.Ended up developing of the atomic bomb,or atomic energy in general.In other words when you're liberatingenergy like this, and if you think aboutit, because c squared is such a bignumber.It takes just a little bit of mass if youcan liberate the energy inside there, you


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get a lot of like kinetic energy and heatenergy and other forms of energy out ofthat.In fact just to give you a glimpse of it,we all sort of have a general idea of thedestructive power of an atomic bomb.It doesn't take a small bomb to level acity and things like hydrogen bombs.Which are based on nuclear fusionactually have even more power than thatIn terms of more peaceful uses of it, forexample.If, somehow, you could liberate all the,the energy contained in a mass of, ofjust three kilograms, okay?So that's, you know, not, not much massthere.if you could do that, if you could turnthat all into energy.You could power a city with a 100,000inhabitants for a hundred years.A city of that size 100,000 inhabitantsneeds a generating station, an electricalgenerating station of about 100

megawatts, a 100 million watts.And therefore just three kilograms ofmatter.Again if you could liberate all theenergy inside there, will power up a cityfor up to 100 years.Now nuclear fission nuclear fissionyou're just liberating just tiny amountsof the mass energy available.But your getting a lot of energy out ofit even in those cases.Now you might ask going back to Einstein.what was his role in especially the

atomic bomb, because you may know that hewrote a famous letter to presidentRoosevelt.Franklin Roosevelt in the United Statesalerting the president to the fact thatscientists had recently discoverednuclear fission.Atually so this was in the early 1930'sright around 1930.And the possibility of either acontrolled nuclear reaction, or even anuncontrolled nuclear reaction usingnuclear fission.

That would release incredible amounts ofenergy in a, in a bomb, was feasible.So, beyond that, though, Einstein reallydidn't play much role in, in thedevelopment of it.he, he was asked to write the letter bysome of, the other scientists involvedwho were very concerned about this.And because his name had cache as itwill, as it were he could get the


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attention that the powers that, that be.They approached him to, to write theletter.Beyond that letter he really played norole at all in the development of theatomic bomb and the Manhattan Projectduring World War Two in the UnitedStates.So that's Einstein's role in that.Certainly in a sense it all does go backto the E equals mc squared equation.The idea that there's an incredibleamount of energy stored in regular,ordinary matter, if we can liberate itand, in some sense there.one other point to make here, about Eequals mc squared, gamma mc squared, andthe like.Is that, we've talked before, aboutinvariance, and the fact that really, abetter name for perhaps the, the specialtheory of relativity would beNot the theory of relativity but thetheory of invariance.

Because one of the key invariantquantities is the speed of light.or, actually, technically, it's the speedof massless particles.But we'll just say, the speed of light,invariant.No matter how you're moving with respectto somebody else in respect to a lightbeam.You will always measure the speed oflight to be, to have the value, c.So we talk about invariance.And turns out that in 1918, so just a few

years after the miracle year of 1905.And even just a couple years afterEinstein introduces general theory ofrelativity.A mathematician named Emmy Noether, whomEinstein considered one of the greatestmathematicians ever.She published a very famous theorem, thatessentially ties in, tied invariance into this idea of conservation of, ofenergy.And she, she was able to show that otherquantities, like if you think about just

if we move from here to there, thatdoesn't change the laws of physics.That's called translational invariance.So we can talk about translationalinvariance.That if I move from here to there, I dothe same experiment.If I do an experiment here, I do anexperiment there.I should, you know, shouldn't get


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anything in terms of my answers, assumingeverything else is equal there.In other words, where you do theexperiment in the universe shouldn'tmatter.Again assuming the other conditions areequivalent.So that's translational invariance andout of that Noether showed that theconcept of conservational momentum cameout of that.Again Momentum is beyond our course butjust the idea of translational invarianceis connected up with this idea ofconservation of some quantity.This case, conservation of momentum.Another idea is rotational invariance.If I, as I turn from side to side, orpoint in that direction, versus thatdirection, or maybe versus thatdirection.I should get the same results.So that I have rotational invariance ofthe laws of physics, in some very general

sense.And that leads to the concept ofconservation of angular momentum.In other words, in order to show thatthese were equivalent concepts.If you have rotational invariance, youget conservation angular momentum.And then, finally, back to theconservation of energy, to show if youhave invariance through time, you know,time like invariance in a sense.That connects directly with the idea ofthe conservation of energy.

So, if the laws of physics are the samethrough time and of course time and spacetime are the key concepts underlying thespecial theory of relativity.And later the general theory ofrelativity.Noether showed that this the concept oftime and moving through time.Whether I do an experiment now or tenminutes from now or two years from nowAgain, other things being equal, thatleads directly to energy beingconserved., okay?

So, in a really deep sense, this allhangs together.If you really start pushing it down inthe depths of some of these concepts.You see that we're not just talking aboutyou know, sort of time dilation andthings like that.But we're, we're talking about thefoundations of Of the universe itself,and how it's put together, and, and how


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it works.So, those are just a few words about,again, right, The Famous Equation, mostfamous equation, E equals mc squared.Again, the, the real form of it comingout of relativity is E equals gamma mcsquared.And we showed that out of that you cansort of get the, the regular form ofkinetic energy.But you really, then learn about thisconcept, or see this concept that, thatmass itself has energy stored in it.That mass, matter, and energy areequivalent in some sense and can beturned from one form into another.And then in cases in like nuclear fissionIn nuclear fusion release huge amounts ofenergy potentially.

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9 - 3 - The Famous Equation (18-29, High-Def) (1)