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Welcome back I'm going to start were reviewing whether Cover the last lecture River comparing the air aired this simple harmonic motion tool of free-fall mulch and I introduce

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We had potential riches for rapid was simple harmonic motion and linear For the year of the free-form For the simple harmonic motion the forces Proportional a position that is I have some equilibrium position at the euro and the force As measured our proportional to the deviation from the equilibrium and From Newton's law as equals and they can then derived this differentially rich Which relates to 2nd Time derivative of the deviation from the And we define this coefficient here OK over there the angular frequency an on gray and the result is motion looks like a locals signed as shown he functional time this time this is position and it also this Equilibrium position not and I wasn't defined have told of the observation which is the the maximum deviation from and a period which is a good distance from our Crestor pressed or not And then there are only equals to hide times frequency Or too high divided by fury The frequency is inversely related So mathematically we describe this As a X equals X. not plant 8 cos I'm honored T plus some angle on a day The ferocity This is the 1st derivative of the coastline becomes a minus sign and a final doesn't Come on what we are a real honor guard plans for the and I get the equals minus 8 on sign on fine for the maximum velocity is the amplitude Time frequent and them untenable the same game again and get the acceleration and assign becomes again call sign and A maximum acceleration minus 8 times on the square on minus 8 times came over 110 then I assume initial conditions at that time equals euro there's some initial position X Some initial velocity S and that then simply air Arab jargon in those Operations We get That they call fine must be equal to access minus X not a signed find must be equal to minors years Long ago And beastly equations are sufficient to determine find a principal Door that 1 uses a little trick when exploits said cosigned square of fine plus sign fine

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Imports while on his way up the 1st equation

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And the 2nd to Croatian than me get this Relation at a square equals access minus not where you have a on and the division the ratio of mostly creations relates the tangent of angle to excess On the go so so much Was that as far as we review Wisconsin and you have a question

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So I repair work we didnt column last time This is the energy of the Earth's simple harmonic motion and 1 off 1 component of the energy we already have the potential and which was this quadratic functions just Racist A quadratic function Potential energy and Bambi kinetic energy off simple harmonic motion is just like

05:13

Kinetic energy always it is one-half times the mass times velocity and so are for the potential energy we get One-half Kate X minus 6 months and bargain and that these formula for X minus Not just goes on and he finds a gets them one-half Kate a spare Cos Times Square on And so for me there Connecticut energy that is one-half And times velocities squared Which 1 half and based on Just told around and getting a velocity the cosine terms signed on That's transported all So

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Our pickles Jerry Kirk's Qantas X to be purchased due to achieve Omega TV plus A fine So this is factors a time on North Korea

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It's clear Because the velocity squared And since we know that ominous square Scale and that means that At times ominous there must be paid and so I get one-half case there signs on that which looks very similar be a potential energy except the cosine is replaced by sign so our monk sort of follow these simple harmonic motion and look Like in scraps and look at how the interplay happened Between potential and kinetic energy And for example he'll be particle is addressed the the mosque where the rest 0 so there's no kinetic energy and it's all potential energy then and the mosque gets least their travels to X includes your across the Europe There's no potential energy at all And the energy is all kinetic energy Equals euro and Kate become racks and then the badly muscles down and reaches the opposite point velocity again euro and you have the same situation here and then I pitched It travels back in for example some distance sex and there's some potential energy from of cost total energy use some of our potential and kinetic energy and using again this relations signs goes on science equal upon ate Don't The equations you just end up to one-half case And that And happens to be the same as 1 half last time Max Show attend the maximum velocity 18 times An armored and of course there Katie equals and on school Dish So I reiterate that we can relate That faded common tea and so on and as angular velocity and the projection of a new form of emotion is a simple harmonic is so so far have represented all are simple harmonic most by reducing springs and a Moslem sprained right can actually do something far simpler here for example on a little kit of suspended like this

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This Friday Sort of deflected a little bit Thomas a deflection is small The F. B. projection on this on the floor It is actually a simple harmonic motion just like for the spring

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So if we make a sketch Here is a great that the strain that has some some length And deviate deviated from the vertical position by small and will fade So because this is a perpendicular triangle The X Old times the size of this And now they're doing on another little Maastricht Leo CO

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That Sein Fein is approximately equal to 5 This is a fairly good approximation as as fight will be And by the way 1 can make a similar approximation of the cosine of the angle and of the same quality America is equals 1 2 on Aug outlook the approximation with this approximation Yukon also seemed to look at how closes the call sign of the city mangled still if it if don't make this approximation matter be tangential force which is restoring force formed this direction spits out from the force of gravity for priority

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And This is the restoring force and this is the force that suspend or the from from the strings that sort of keeps straight talk so this is the call sign off 8 of the same and faded from view this and that we don't need to Loria about this 1 as long as it's not Big enough to wear a tear apart the street So what be interesting 1 a tangential for time signed favor Walker and I make this approximation Ng a times space And as As for the Mosul it has a minus sign So bear and the minus side is because production restores the mast Equilibrium position bottom so West Sims There F tangential equals miners and over times X there since calling this an X which is a wild times and to see situation he misses simply do the definition of an angle that remind you of about the way the mathematicians find an angle is their maker unit circle

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And this is the angle fado and they say This distance here and that is the amount of radiance for that Anger so when you go all the way around it's Hi and that therefore the fate of Simply this distance X divided by Wonsan scissors You wounded so-called more general X divided by the way so if I look at the situation F tangentially

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Approximately equal miners and Gino well-timed X Then it looks like A reports for the thing Where this spring constant as NGO weekend Basically use work we had for except We replace uh the case with Instead of on us where are equal scaled and Becomes simply dual world can't got so let's look at my armor debt So sort of remember the these the period After strength the period becomes much faster Jim And actually 1 could fund could use such a simple device to measure the acceleration of gravity because all I have to do is measuring the The length of the spend and because her This equation of ready of the the frequency with which it thinks it tells me an biggest G So There's also a different type of friend and for this And we have assumed that all the Masters concentrated And at the bottom so we assume that the the string doesn't contradict when doesn't have to bleed make this assumption can take any ot shaped object and It becomes a pendulum in the same way And it also seems to be a simple harmonic more that So our 1st of all my apologies for computer font problem what I meant to say is that uh the pork Called the boss Times the level of the divided on correctly

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So this is actually a rectory creature

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But if you don't like crossed products like most people do then that it can simply Calculate gonna be perpendicular distance take into account this Angle favor and Then the talk actually becomes The force times this perpendicular sold this perpendicular distances H. time a sign of angle fade So at age is the distance between send-off Gravity And A key point of suspension by the way 1 can you get us that treat me baseball bat as a pen and in fact from on published different Played baseball efficiently Now we are doing the same trick for the year What An obnoxious sign favor the from we simply assume that favors small and therefore the Comes and G H simply times No use Newton's laws for full rotation which says That the um the talk must be be equal to the moment of inertia times the 2nd derivative of the angle And if we do that because of this equation big again get the equation of simple harmonic motion 2nd time derivative equals some factor times a quantity itself And If we define this quantity of which is the moment of inertia divided by the Marseille times this Distance h And I will call this the reduced length will become apparent why the moment because now I can't take this this equation here and I can write it As these could played a cheese Korea equals mind she overall star favor so now the equation looks exactly the same as the date for me and for the simple and the only difference this uh that instead of a simple length I have this a reduced length which is not so easy to determine the reduced length was not simply the distance so I can't let that take the physical Pender moment find the scent of Marseille and say This is my reduced length it actually depends not only on the shape of

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The object but also on how the Masters distribute so if this part is modems than that Pomp and that will change however quantitatively The fact remains that It's swing slowly for a long reduced length check and strings more quickly for uh slaughter which now I that this simple kind alone we can use To measure the acceleration of gravity Assuming they the acceleration of gravity We can turn it around and Was A pendulum to measure the uh be reduced length and because of this formula as moment of inertia and blues and an example so

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This is Provides actually an easy way to measure the moment of Russia religion matters is not a very easy quantitative measure general so we can measure the of 1st of all there any axis learned rich If we simply have reverted to a wrong desire On and then measure the period of observation on the acts So let's suppose for example that they have 1 kilogram stick and that 1 kilogram stick is balanced and appointed this 44 sending leaders from 1 end so we know that the difference between pivot and and of must and we also know that it also late With period of 1 . 6 seconds when the air The granulated swing on it Madam Arab What is the moment of A mirage The acts and the answer is you know that bombing on equals to high divided by period And if you drop in the number of We want France acceptance as a good team Sondheim 2 per 2nd or on Americans gray And we get 15 . 4 4 per 2nd square We can then use the formula reduced length And turn it around So Amadeus Carrera Nicholas P. G O L staff that means that I will start will go on and scored recently

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By both sides of the equation the fact So if do that then we can calculate the reduced length

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. 6 3 5 G We can then use the formula for the year that relates moment of inertia and reduced length Same battle calculated what is the moment of inertia so this is the year 42 sending leaders or . 0 . 4 2 meters times B . 6 3 5 needles and It it and then it is 1 kilogram of Muslim pilgrims to do that and you get the actual Burke So all our fault so I guess that a question So That's because of the car OK I'll explain that attended school back

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To Still ash just to make sure everybody Asked me to explain the concept concept of reduced length forget the ideas That we want treat this complex handover like a simple 1 And since the Eco Asians are are similar 1 equation as had a T-square were actually in this story

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X equals about My Jean All X and in the other cases please These growth figure the team's career equals our mind is a and G H or what I Times state therefore I want But to define a quantity of 4 Both of these expressions have G and so are their fathers nice but The remainder be crammed age I want a single just like a like a legs so I know from those 2 equations as enraged although I must behave like 1 always length and now I am since this is not really length It's just a mathematical expression I say this A reduced length And the physics we actually do often such as such Fix will be I don't be fined for formal equivalence of various problems And then be defined sectors Quantities like this do that move on to damped harmonic more

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And a lot of considered complicated unfortunately a and I take must And I waited long enough Especially if there are some friction involved Power and eventually the mold stops And called damped harmonic motion it is no longer a simple harmonic emulsion and it is mathematically complicate it depends of course how we assume the friction to behave 1 A case that actually treatable He added that thing for us is proportional to the velocity that means the faster the object move moves had dampened So I since Since you know the velocity is the 1st derivative Next month It's not then that we simply have this coefficient beat her She called the damping caught fish So now badly this corrasion this type of situation have becomes more complex because it's not only the 2nd derivative equals minus some factors times the quantity itself but has also this at that being forced to consider and that leads to this A term that is proportional to the And of course the way you get This equation tires at equal cemetery for Seoul our new still holds the only difference was that Mao OF is the restoring force Class leader the damn thing for us

29:06

And if I divide this equation 5 months and then Arab I get the acceleration and I come back to this creation he

29:25

So this is a nasty and differential equation because we can't Easily guess the answer answers we did for the simple harmonic And 1 possibility is to just normally and and then a a a calculus and prove The case but there's actually a somewhat more elegant way if users complex variables so that we make it clear that I don't expect to be able to do complex variables In the exam or even a new home so I'm just going to explain it in terms of complex variables because their complex variables each some complicated by actually not all that the idea of a complex variable is that we have The We'll father just imagine And we assume there's a quantity of Iife Richmond that square but it equals miners And because we call the 1st part of The real miners down and the 2nd part the I recalled imaginary Omega Prime and both There are just normal real number and as such are complex which is where the wrote complex variables comes from 1 can calculate just the the usual Addition and multiplication and longer justice for example on the show in the 2nd line they fire multiply X 1 plus I might want times X to write to their need to multiply this that and And the crossed her arms and my death Next 1 it's still plus ice their wire 1 and then the 2 mixed terms and since I'm not that I ready equals close-minded 1 by assumption I get this expression that for example is exactly how Occasional tool complex variables work and of costly edition is even simpler than the multiplication because they would can simply drop the practice and since it's just a plus sign relates the complex However the real trick it helps us for this problem is a so-called boiler relation Which is already several centuries old This is not cutting edge mathematics where the autocorrelation this for mathematics about what Equals MC Square physics Delivery import and the order relation as an are not going to want to even tell you where possible that this fool just take my word for it it's about He said the ionic got for any bombing this call sign I signed on In particular it got equals to high then call assigned to highest want and signed Kind of Europe followed Each Ito the iPod time equals what So if we are using all complex variables and say That's 4 x minus eggs your guessing the answer each with a complex variable time time A broker and said about my best I will receive cheating

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He took virus Gallup team times cosigned team classed by cyanide ominous so a wrist Hottest sort of what we want is that part or that part so if we actually wanna related to something physical we take the real Ponte call and since this is a real number this is simply Ito the miners come up to co-sign aren't so the way this function looks like we will ever come back to this next section just on a show you the way this function

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This is an exponential function exponential function looks roughly like this

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Spiegel and be multiplied as exponential function is kind of like the Abbott told officals assigned So it's lot of roast like this and that he has drawn it more and more beautiful lead with the rule

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Cut And it looks like that And these other solution For the same equation and I will tell you about it next