all right greg simple harmonic motion okay now you have uh in principle mastered the equations of motion right two column table one column table depends on what we're doing right but and and you have equations that you love i know you love them right because you you use them a lot right uh x equals x zero plus v zero t plus one half eight t squared but those only work if you have constant acceleration right so if you're sliding down the ramp and you have variable friction they don't work if you have um if you have projectile motion but it's got a rocket on it doesn't work right it has to be constant acceleration for those to work now that means up to now whenever we've looked at springs yes springs don't have constant force do they no and if caitlyn if my spring doesn't have constant force f equals m a constant acceleration no so can you use those equations for anything involving a spring no because springs don't have projectile motion they do this right they flop back and forth which is in code simple harmonic motion simple because we're going to use ideal springs harmonic is a code word for sine wave cosine wave and motion well we're going to figure out how does that how does that mass on the end of a spring move even though we don't have constant acceleration okay now that will mean you now have two sets of motion equations and in any particular problem you'll have to decide whether to use the first set or the second set big clue does it have a spring in it okay because if it's got a spring you want to go to these right so that's see what else okay now uh kimberly i i read the text last night just to make sure that i would cover the appropriate things and as always i was just astounded by the sheer number of little itty-bitty equations that show up when we talk about simple harmonic motion okay there's no way around it okay but i i will try to bring some order to the chaos of all the equations that are in there although this probably on your formula sheet deserves a corner of it with a box around it where they're all grouped together okay and you might as well start writing them down as you do the homework all right now in general lily the two big things we do with simple harmonic oscillators are masses on springs and the small angle pendulum you'll do the small angle pendulum in lab okay you'll look at it uh there are others but these are going to be the majority okay so well i said you know a big clue is do you see a spring oh or do you see a pendulum right because those are the two largest places where these occur and emily we still believe right f equals m a but we already know the force from a spring don't we right it's minus kx so what we're dealing with here is acceleration equals minus kx and you notice it is manifestly not constant just in case you missed it in my first words right it it depends on where you are as to what the acceleration is right if x equals zero well there's actually no acceleration if x is at an extreme value then the acceleration will be the largest for the motion and it just varies back and forth as the mass oscillates back and forth or the pendulum ticks back and forth yeah okay uh yup the equations of motion don't work here and just so that we're clear simple harmonic motion or simple harmonic oscillators i kind of float back and forth between them is periodic it has a period right it goes over here sometime later it's back over here then it moves away now it's back over there right it's periodic like sines and cosines are so all simple harmonic oscillators are periodic and oscillatory but not every oscillatory motion is simple harmonic so we have really focused in brooke have you ever seen this weird phenomena when the wind blows and you'll get a street sign that does this okay it's periodic right it's oscillatory but it's almost certainly not simple harmonic motion right the the physics behind what makes that sign do this when the wind blows on it is is really complicated if you think right the wind's blowing on it it's it's not moving back and forth as though it's twisting right what what makes it twist very it complicated physics behind that okay it's actually related to why airplane wings have that little curvy thing on the end okay same thing all right so in general mackenzie we have three general characteristics when we talk about simple harmonic motion we have the amplitude of the motion hey is it vibrating like this or is it swinging back and forth through meters okay that's how how big is the swing is the amplitude we'll get a precise definition as we go along okay the period measured in seconds the period is how long does it take to go from one point through a full swing back to that point okay and we have the frequency which is related to the period and now just to be very concrete about these things austin uh the amplitude is half the peak to peak in other words if it starts at a meter moves to zero stretch and moves off to the other side a meter the amplitude is one meter not two right it yes i understand the entire motion the peak to peak the extremes of the motion was two meters but that's not the definition of the amplitude the amplitude is half of that okay if you care it's it's the it's the it's the distance from the equilibrium position to the extreme displacement you have no idea how many people get very frustrated doing expert ta because they read the problem or they look at the little graph and the problem shows that the peak to peak is 20 centimeters and so you use 20 centimeters in all your calculations not recognizing that oh wait a minute amplitude is half that okay don't do that at two in the morning all right okay um charles robin yes okay um for the period you know we're talking about something where as a function of time we have an x and it goes thusly yes okay that is the period we're going to call it big t it's from top to top could it could we measure it bottom to bottom sure can we measure it from a zero to a zero well yes but you have to be careful right because you would have to measure say from this zero from this one to that one right because in one case it's going down the second one is different because it's now going up and then it's going down you always have to go through one full cycle if you start up you can measure down and up you measure down down up down if you measure from zero you got to go up through zero down and then back to zero again it's a two in the morning thing where you misinterpret that and then life doesn't go well ian i'm sorry okay i'll just say it right up front there are actually two frequencies i'm sorry you're going to see later on when we start getting some of the equations not only are there two frequencies there are two omegas right and they're different and they show up in one equation right and they might they look identical right and you just have to deal with it i'm right all i can do is say i'm sorry it's a little bit like you know i read the red book you know exactly what that means right you have to interpret the equations in context you can't just go oh hey oh that letter okay but uh how many cycles per second okay so you watch it go one two three four five and you time how long it took to do five or ten or one and that tells you how many cycles did it do right and that's the frequency okay f uh it's measured in hertz right or one over seconds they're equivalent uh but there's also the angular frequency uh and this is not unrelated to the angular velocity that we had before it's got the same letter in fact uh you'll see it this is actually omega but it's not the same omega that you had for rotating objects because this object's not rotating it's oscillating sorry okay okay you just have to work at getting it straight right but the angular frequency is 2 pi times the regular frequency this would be measure in radians per second okay all right now i'm not going to make you get them out but at some point you need to get your calculator out and figure out how to put it in radian mode because we're now entering that unfortunate part of the semester where if you forget to put it in radian mode none of this works okay and i mean it's not like it's a little off it's just totally screwed up okay because we're going to be taking sines and cosines of radians not just not just angles uh shelby yeah not only that but the final exam is cumulative so you'll do some problems where it's going to ask you about the direction of the force in degrees and you'll do some problems where you're going to do oscillatory motion and it'll be in radians and please don't forget to switch back and forth okay that never goes well all right again i'm sorry okay that's kind of the theme of the day here okay adam yeah we start with energy because energy is so fundamental so i have this spring going back and forth yes and we've got a little cryptic movie here where it goes a b c d e right but it starts out in an extreme and then at the extreme where v equals zero it has no kinetic energy it's all potential it's all spring potential right then at at the middle where x equals zero it's all kinetic and no potential and then it reaches the extreme in the other direction and now once again it's no kinetic all potential and it the spring this is a continuous exercise in transferring from kinetic to potential to kinetic the potential and in between the frames here there are places where it's got sum of each yes but we write it out one-half mv squared equals one-half kx squared is some constant okay and what is that constant well it depends on how much you stretch it right if i were to take bailey if i were to take that spring and really stretch it there's going to be a lot more potential energy right so then i would expect in the middle there'll be a lot more kinetic if i only pull it a little bit right then the constant will be smaller right but you know i can always find the constant because i know i know there are two very special places right one special place for the spring is to say hey when x equals zero i'm going to have the maximum velocity i will have no potential energy and i could find the constant if you tell me the velocity at the middle i know how to find the total energy oh or if you tell me hey i'm at the maximum position then there's no kinetic because it's not moving and if you tell me how much it's been stretched at the maximum i can find that constant because it just means finding the potential energy right so i've got that and so then we end up in fact that hey in the middle one half mv squared max must equal the energy when it's fully stretched is one half k a squared and so i can actually relate things like the maximum velocity to how much it stretched yeah because a is what is x max i like a a little better than the notation in our textbook which is a big x partly because i learned it as a and if i don't write a i get really confused okay so your young plastic brains can handle two different things i'm stuck with it right you'll notice christina uh we got no motion here yet do we we really haven't done what we did way back when with constant gravity where we actually derived and said hey if a equals a constant then i can find v and if i find v i can find x and we work those equations out yeah we haven't done that yet it's hard okay it's so hard that as with projector we're going to start with the acceleration f equals m a right and that f equals minus kx and so i i i have my a right and i know how a relates to x and at this point you'll notice what it says with help from a friendly mathematician okay because we we need some we need somebody that's mastered calculus okay but if we find somebody that's mastered calculus they will tell us well if you are willing to say i'm only going to worry about cases where i'm going to pull it to the side and let it go starting from rest and where i've pulled it in the positive direction they will tell you that oh a that x equals cosine omega t v equals minus v max sine omega t and a equals minus omega squared a cosine omega t and you'll go yeah thank you very much and if you plotted those right you'd get those three nice curves up there the first one is a cosine curve we can tell that because it starts at a maximum and t equals zero and we know the cosine is zero uh is one when the time equals zero and we can see that it's got some amplitude and of course if i pulled it farther that that cosine in the very top would just get taller right okay emma we we can look at here and we and say well omega how do i know what the frequency is going to be well our friendly mathematician would tell us in order for that to work with what we had up above of a equals minus k over m times x omega it can't be anything we want it's fixed and it's fixed by how stiff is the spring and how big is the mass now that kind of makes sense if i have a really stiff spring right i mean really high spring constant it's likely it's going to go right if it's a sloppy spring and really loose and it doesn't take much it's going to tend to go okay it's going to go over here and then it's going to pull back over here and it's going to be slow and the same thing it's 1 over m right if i've got a heavy mass it's going to be hard to make that go back and forth fast right because you're accelerating this thing all the time right it it when you get out to the to the ends you got to stop it and turn it around and make it go the other way right if it's a really big mass it's going to be hard to do that and so we see oh if the mass is big it's 1 over so the omega will be small right the frequency will be quite small if if if k is large then omega is going to be large right now intuitively you might be happier with f okay hope you're happy with f because f is a nice symbol omega is this cryptic greek thing right so omega must be more complicated than f you know there's just a relationship of 2 pi right we had that on the earlier slide 2 pi f equals omega you're going to write that in the corner box on the formula sheet so you don't forget it but we depending on the particular problem you might be told the frequency you might be told the angular frequency i'm going to write these out in terms of omega just because it it takes a while to put these formulas on the screen right and it's easier to put one greek letter than it is to go 2 pi f i could right there's no reason i couldn't rewrite these as 2 pi f everywhere through there if you're more comfortable with that that's fine okay but but in any inst in any case we look at that x and we say oh if omega is big right if the frequency is big let's say oh yeah if the frequency is big the period is small yes because i don't know i'm going to write it on here maybe it was written on an earlier one the period is one is one over the frequency i could write the whole thing in terms of that too all depends on what the information is but if i have a big f i have a small t high frequency short right if i have a high omega this means it's going up and down really fast right so the period will be short if i have a um small omega oh the period gets stretched out and as much as some of you have heard me talk about looking at your homework and say oh well you're just formula stuffing uh this kind of lends itself to formula stuffing okay in all honesty yeah um yeah oh yes yes addison yeah there we go um yeah and we look at the first one do you see the cosine in there right it starts at one and it oscillates then the velocity starts at zero well yeah we said this is the answer if you pull it to the side and let it go from rest so it better start with zero but if you notice too if i pull it to the positive direction and let go what direction does it move it moves negatively right it has a negative velocity so we see that in the picture there right and then oh the acceleration well the acceleration is biggest at the ends because you're stopping it turning it around but two the acceleration is a negative at the beginning okay but oh my goodness except for the size the position is this the negative the of the acceleration can you see that right so those general characteristics will often help you interpret words in the problems where you know we try to write them in a way that fool you right so say something about when the acceleration is a maximum find this well think back to this right when is the acceleration max and look at oh when the acceleration is maximum the velocity is zero but the position is at the extreme negative value that that little picture will help you translate words okay okay that's good unless there are questions everybody's stunned yes you want me to go back no not yet the different omega comes when we get the pendulum that's the that's the same thing that's in the inside yep good question did everybody get it we don't have the the two different omegas in the formula yet so the minus omega squared cosine omega t same values same numbers okay both of those are 2 pi f you look confused i'm gonna i'm gonna back up here just a minute it's worth uh getting this i can also go here and go four pi squared f squared no why did it do that all right we gotta do that we got to keep we want to do that does it show now we very carefully do that and let's see if we can turn this back on yes all right those two omegas are the same that was the point now you said neither none of neither no omega in there is angular velocity it's angular frequency there's gonna be yes there will okay in the very last module when we do pendulum it's gonna get weird okay and it'll be perfect because by that time you will have totally tuned out okay and then when you get to the homework it'll be important and it'll be like i don't get any of this right what's that yes yes i will