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Professor Walter Lewin Hooke's Law, Pendulums, Simple Harmonic Motion, Mass-Spring System

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MIT Porfessor Walter Lewin's lecture about Simple Harmonic Motion, Period of a Pendulum, Period of a mass-spring system, and Hooke's Law

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today we're going to talk about  Springs about pendulums and about simple harmonic oscillator picks in 81  if I have a spring this is the relaxed length of this spring I called a taxi to 0 and I extend the string the spring then there is the force that's wants to drive this spring back to equilibrium adolescent experimental facts many Springs we called an ideal springs for many Springs this force is proportional to the displacement X so this is X if you make x 3 x larger that's restoring force is 3 times larger this is a one-dimensional problem so to avoid to Vector notation we can simply saying that the force therefore is minus system constant which sprinkles Disney Springs and the spring constant has units Newtons per meter kind of signs takes care of the directions Barb action restoring Force FX holes as wood floor how can we measure the spring constant difficult  I can use gravity  serious the spring it is relaxed situation I hang on this a mass m and I make use of the fact that gravity exerts a force on the spring and Wendy's new equilibrium position spring force of course must be exactly the same and so I could not make with my ex and I could have here I can change the data points which scatter around a straight line ND spring constant solos then if you take if you call this Delta  can you call these Delta X then the spring constant K is Delta s / so you can even measure it at this point where the spring is restart when the spring is ready under tension you'll be surprised how many Springs really behave very nicely according to Hoops law I have one here AliExpress resmart 13 centimeters apart and everytime that I put 1 kilograms that it goes down it goes down this March I put another kilograms goes down to this Mark I put another kid my mom and it goes back to  all the way down if I take them all off so what I've done is I effectively when's the longest truth and if I take them if it isn't ideal Springs back  I went to does requirement of course Real Spring according to who sings to the point that is no longer behave like Reformation look that's easy  for sure who flow is no longer look how much longer this ring is in this before a limit how far you can go before you permanently deform the spring what I have done I wind up along the straight line and then something like this must have happened when I relaxed when I took my voice off this plane was longer at the ends that it was at the beginning so I haven't met extension which we will be there very nice of course to do that to discipline there are ways that you can also measure the spring constant in a dynamic which is actually very interesting I have here spring  and this spring this is Exit Zero Chipettes now to display mass and it has to be on a frictionless surface and you will see when I extend it for the distance  that you get your force in Spring force that drives it back we have of course gravity and and we have the normal force from the surface isn't in the right direction that is no acceleration so I don't have to worry about the forces Direction  if I nothing also right I will start to us about this point back and forth then I will show you know you will find that the period of oscillation the time for one of the mass / the spring constant k then you can calculate K if you can calculate the mass can use gravity here to measure a mass that you see the time that it takes for this object to also made ones back and forth is completely independent of how far I move it out which is plain old into it this but you will see that comes out of the depends on how far I also like this ocellated like this as long as you will see that the period is independence of what we call that I'm going to drive the situation now for when I do case there's no friction has negligible Mass compared to this one let's call it a massless spring  so now I'm going to ride down Newton's second law any which is only the X Direction equals - KX  is the second derivative of position for which I will write X double dots MX equal I / MX double belts with k m x x equal 0 and this is the most important equation in all of physics  differential equations  some of you may already have sold correct solution for X as a function of time you substitute back into what would a solution be to this differential equation I'm going to make you see this oscillation X as a function of time and I'm going to do it in the following way I have spray paint which is suspended screen to Springs and I can also late it thoroughly which is ux Direction like that  so X changes which time I will introduce by when the spray paint is going to spray don't you just secretly please what kind of a weird curve as a function of time can you tell me some message because I will try to get spray paint okay  alright did you give me a hint could you please I will cut it here and then you  Doterra cancel because it is messy when I put it to take it out this way can you form back just walk the Great I hold up to talk so if you can see it's fine  okay what is what is remind you of dinosaurs alright let's try it to substitute in that equation a sinus or a cosine solution whichever one you what makes no difference  so I'm going to substitute in this equation this my trial functions that acts as a function of time is it count minutes * cosine this week all the empty tubes notice the cosine function is the highest value is + 1 + the lowest value is -1 so the answer to indicates that is the farthest displacement from zero on this side this Omega call the angular frequency  don't confuse it with angular velocity recorded angular frequency and the units are the same the units are in radians per second the same as end of the velocity  if I add phone this time Little T if I had bombs that by 2 Pi / Omega if I plant This Time by 2pi / Omega then this angle increases by 2 points what's the time that it takes 40 oscillation to repeat itself  and that is in seconds can you can determine if you want so you can Define The Freaks and switch do we have what we call the phase angle  and I will return to that sexy Radiance and this trial function I'm going to substitute now into this equation to the first thing I have to do as a function off time  well that's my function I have here first for 3:30 that becomes -8 Omega Omega till I get to signs of Omega t + 5  of course I could have started off here with a sine curve I hope you realize that I just picked the cosine accidental.  No I got another Omega out so I can - 8 Omega scratch the delivery what's the sign is the cosine cosine Omega t- a cosine  so now I'm ready to substitute this results into that differential equation  this must always hold for any value of x for any moment in time and therefore the only way that this can work if only and therefore we now have the solution do this problem so we have omega truth of MLK Angora striking remarkable that this is independent of the what's the truth and it's also independent of this angle side angle business angle well you can think about the physics actually when I start this oscillation I have a choice of two things I can start it off at a certain position which I can choose I can give it a certain displacement from zero and Subaru but I can also when I let it go give it a certain velocity that's my choice so I have two choices where I let it go and velocity I give it and that is reflected in my Solutions skip the result of a and the result of which conditions I want to do an example by you see how 8 and follow from the initial conditions Arby's the objects equilibrium velocity which is -3 meters per second equals 10 Newtons 4 meters and the mass of the object is 0.1 kilogram  and now I can ask you now is X as a function of time including the answer to Days Inn the phase angle  well the first calculate Omega what is the square root of K of M that would be 10 radians per seconds  Superior C  which is 2 Pi / Omega Russell  6.28 seconds  and the frequency F would be about run 6 some numbers this is not my say This is 40  6 - 8 + 1.6  Supai you can 6/10 alright so now I know that equal 0 so I see my solution right there  right here in 00 0 equals 8 times the cosine of 5 well zero if I release it is 3 Pi 3 pi over 2 condition that the velocity is -3 equals -8 + 8 we don't know yet and there we have omega squid oh my God sorry which is 10  t0 I get the sign of Pi if I pick immediately I told a solution which includes now fired is at x equals plus 0.3  time is the cosine of Omega which is 10 plus 5/2 so you see that initial conditions what the conditions are they determine my age and they determine life define Yoda found a minus sign here and that's exactly the same so you would have found nothing different I want to demonstrate to you that the period of oscillations Nolan to Independence of the amplitude that I give you off yet and I won't do that to you is this track  is objects 180 6 + 1 - 1 G call M1  I'm going to ocellated and we going to measure the periods what instead of measuring I'm going to measure and I got an error which is my reaction which is about the tens of the seconds  that's about to reactionary that we all have roughly we got $0.02 and we get again plus or minus 1 points I will see how much do they differ they should be the same existence I'm going to give how many centimeter offset switches and then I will start it when it comes back here when I start it  rum two three four 567 h $0.09 what 15.16 15.16 seconds spring constant from this now cuz you know the mass  now I'm going to give it a displacement and amplitude which is twice as high try to make it 40 centimeters 40 centimeters is huge displacements one  two three 4 five six H9 10 1541 Street fantastic agreements with my measurements he won't always get that close because my reaction time is really love much better than this  no I will show you something else which is quite interesting in that is how the behavior of the period is on the mass of the object  I have another car which way is roughly the same I'm going to add to the two together and so we get into this about 372 what time is when the person is one comes in because our skin is no more extras and we find this to be turn oh I'm going to measure the pain.  Off this object with mass and twice the mass so that should be the square root  open 2 / m 1 * 10 * 2 periods of until I can make and I will take the square root of 2 and I multiply that by let's take 15.15 and so that comes out to be 21.42  2144 to snow clear that this too is Meaningful and I'll come to 6 solar questions what is the prediction  I miss no becomes a little tricky so what I'm telling you now may confuse you a b**** not meant to be but I really won't hold you responsible for it you may not think that the uncertainty in these measurements photos from the Insurgency in this which is true which is about 0.6 per cent and from the uncertainties sense I go because you may think that you can shoot until you there C equals the square root of 372 + 1 - 1 divided by 86 + 1 you should argue this is roughly 1 out of 200 is about Tennessee to Aris for the points 5% and the reason why that is completely wrong but has to do is the fact that these two errors a couple to each other 186  the best way I can show you this suppose I measured and 1/1 which would be does an 86 + -1 / hundred and 86th Plaza - 1 object you / the same object if you would say so I will know you was that I don't know who is correctly and you take the next percent that the error in this ratio is really much less than .2% you can almost forget about it in the final answer and so I stick to my predictions and are we going to get the observation 10 t x + 2 which is double the mass and of course always has my uncertainty of my reaction time so I will put the other mask on top of it come see you  play some together so that they won't that we go so  I hope I get that correctly describes of two times 1541 V one two three Ford 5 6  7 h 9i hope locating 21.36 21 points 364 if you want to see that the agreement is spectacular remove this too of course it's obeys hooke's law and it is also nearly Mathis fearsome ass and he's all set at an angle Theta the length of the pendulum length of the string  is gravity here mg  and the other force on the object the force is attention something suicide with. TJ's extension T in Newton's else the thing is going to Arc around like this and going to oscillate  how cold is in y direction and I call this The xx and x equals 0  well I'm going to decompose the tension in do I add into v action as we have done before  so this is going to be components is the X component components equals 2 cosine Theta next components  equals t sign fader  and now I'm going to ride down to differential equations motion in the X Direction  second Newton's second law equals this is the only force in the X Direction it's a restoring Force just like with the spring and therefore have to give it in minus x minus the sine Theta  see itself easily be a functional so I have to allow for that the sine of theta equal Exposition x / l- with maybe a functional next / El differential equations and I prefer always for this to write down double dogs now Direction direction I have my double. this is my corrections - mg equation equation to we have to solve differential equations which is hopeless. it looks like a zoo and it is a suit  and now we going to make some approximation and the approximation said we will make with you awesome in physics when something also lights is what we call the small angle approximation playing we will not allow fada to become riches in radians equals much much less than one week old at a swollen go if that's the case the cosine of theta very close to Rum you will say wow close to 5 degrees the cosine is 0.9 96 that's close to 110 degrees the cosine is 985 that's only one and a half percent from 1 degrees to do an extremely well so this is number one of the small single approximation Look at video Extrusion that is object colibrium in X erection the exclusion of mixing Direction then the Excursion in the X Direction provided that you angle is small  I'll give you an example  5 degrees is Excursion is only 4% of this excursion is only 9% what's the Excursion in the white erection is so much smaller than the extractions to be roughly zoom in which these two conclusions which turtle from the small angle approximation I go back to my equation number two  and I find that the 0 equals which could be a function of theta cosine Theta is 1 - Angie notice it's no longer even the functional Sada so I simply have in mind that I can make the same it's approximately I substitute in my number one x x double. And I'll bring it on the other side plus x x / L equals 0 no wonderful results X double. plus v l x x equals 0 and this is such a beautiful results that it almost makes me cry is a simple harmonic oscillation looks like carbon copy there  do we have tails in there we have a deal that's there is no difference so you can ride down immediately the solution to this differential equation  ex will be so many times the sign of Omega t + 5 skip the pendulum will be to Pi times the square root of love  just falls into a lamp all the work  I want you to realize that these results for pendulum have their restrictions small ankles and we discuss qualitative how small you would like to allow and also the mass has to be exclusively in here and not in the string we pull that a massless train  forgive you some rough idea of these periods will be substitutes for a meter make for G9 X and what you find is that bomby the long has a period of about two seconds  1 2 3 4 five six to go from here to here is about 1 second  tell four times shorter This Crowd Forest then the periods  the period is changing four times shorter. 25 centimeters trying to be here in seconds one two three 456 that is extremely sensitive to the length sleep  I don't want to compare with you the results that we have for this results from the pendulum to give you some food inside we have the string what time is it  and I'm only going to look at the. See which ear is 2 pi divided by the square root of MLK here is 2pi times the square root of love if I look here there's a mess in here if I look here is independence  why is there a mess in here that's very easy to see and I extend to Spring over certain distance then there's a certain Force  the force is independent of the mass put at the end of the spring the spring doesn't know what the masses you put on hold his nose his lungs and I want to go back to the equilibrium  the force is a if I double the mass that fixed Forest will give on the mask the acceleration the acceleration goes down the period of oscillation goes off so you can immediately see that we split the entry into the. Now go to the pendulum if I double the of my bulb at the end of the pendulum then vertical components of the tension goes from double  tell me this restoring Force which is proportional retention goals so you can simply argue that there should be no mass in his ins  how about this K if K is high that is pretty stiff what does it mean to stiff spring it means that if I gave it to small extension Spring Forest is huge spring Force the acceleration on the given mass will be high if I have a high accelerations the period will be short Siri it will be short imagine that you have a pendulum in outer space if there's no gravity nothing the pendulum we'll sweep the period of the pendulum will be infinitely long going to the shuttle frame of reference perceived they wasteless remember that preceded in the shuttle whenever you take a spring in the shuttle and it does object using the spring on the shuttle and that is also things make sense when you think about it in a rational way we have here in B6 100 oops what is a Pena which is some 541 meters end there is a mass at the end of it which is 15 k cramps the length is 5.18 meters and the uncertainty is about 5 cm we can measure it any better send a message to me. It's about 15 kilograms the. which is 2 pi  x the square root of love if you substitute in your length of 5.1 meters you will 4.57 seconds Four Points 5 Seconds you're going to have a half a percent error in your. So that is about 0.02 seconds so this is my prediction and now I'm going to what's the latest for you and I'm going to do it from two different angles I'm going to do it at 8 10 degree angle in order to get my relative error down I will also late 10 times so I'm going to get maximum  roughly five degrees I get 10 ft equals something plus on mine is my reaction which is 0.14 second time an old .1 seconds Abba movies on way because that 15 kg of hits this that is not funny old lights 0 I have a mark you on the floor this is about 5 degrees this is about 10 degrees I will first who was from 5 degrees one ocean when it comes to a whole Cialis the time of this for me I count on you that comes to Comedy you ready you ready you sure I'm ready to  okay I'll keep calling and don't confuse me again you're completely responsible for the counting so you only have to tell me is when when 809 is coming up that's all I want to know bottom Erie Street don't even bother me before just let me know when I have to get in position for it the final two what is almost no dancing on this Spangles in the church actually see some kind of friction 9 Night Lights 45.70 45.70 where's my truck 45 .70 football prediction rough where are exactly you get the picture that is truck because my accuracy is no better than its ends with e 10 degrees I want to show you the effect on the angle from 5 to 10 kill humans okay again relax and count nerve-wracking terminal 7 did you expect anything else 45  .75 what are the most remarkable things I just mentioned to you that's the periods of the oscillations is independent of the mass of the object if I joined the bulb and I swing done with the bulb thank you should get that same. Or should you not  I'm asking you a question before we do awful experiments  what Superior come out to be the same or not some of you sink it to the same  have you thought about it I'm a little bit taller than his object and that therefore it may be effectively length of the string has become a little less if I sit up like this and if the length of the spring is a little less. Would be a little shorter yeah be prepared for that I'm also I will try to hold my play as hard as I possibly can in order to be at the same level no Zuko this hurts I want to hear your love go turn to was Walter Lewis 45 .6 + physics I'm telling you I'll see you Monday have a good weekend

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Professor Walter Lewin Hooke's Law, Pendulums, Simple Harmonic Motion, Mass-Spring System

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