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Intro to Springs | 24 mins | 0 completed | Learn |

Intro to Oscillation | 31 mins | 0 completed | Learn |

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Example #2

**Transcript**

Alright, so the big equation you worked and some actual stuff, you pull 5 meters from equilibrium, doesn't say which way, doesn't matter and it says, 1.5 seconds to reach its equilibrium position, so if you draw, it's 1.5 second to go from here to here, okay? Every time you do anything with time, time problems in these problems is always going to be, what you really want it's not the time, you always want it in terms of period, right? So what really matters in these problems is period because that's what is in all the equations, not t itself, so as soon as I write this 1.5 it'd be a good idea to know what the actual period is, in that case period would be 6, right, because it's four times, is 1.5, 1.5, 1.5, 1.5, so as soon as you write this, whenever they give you a specific amount of time, you need to get the period out of that because the period is the mos useful measure of time in these problems, okay? So, right away so you don't forget, you don't put 100 later, so you don't make a mistake, right? Like, oh shit like a quarter of the whole thing, alright. So, now I'm going to write the big equation and se can get this right out of here, here is my k that's good, I know m, I know t, and also know 2pi, that's it, all I got to do is focus on these two guys here, and then you got to get the answer, okay? So, 2pi over t equals square of k over m, obviously to get this I have to square both sides to get the m, k out of there, if I do this first and then square both sides, so 2pi over 6, let's get a number, m is 4, let's get number for 2pi over 6 and then we'll square, if you do it the other way it doesn't really matter, this is 1.05, 1.05 and I'm going to square both sides which leaves me with this, 1.05 square, 1.1, so I end up with 1.01, no, 1.1 equals k over 4, so k is what? K is 4.4 newtons per meter, cool? Alright, before we move on I want to talk about something real quick that I haven't mention about this big equation here, okay? This equation is going to solve a lot of problems for you, for example, the very first three equations of you homework, all are coming out of this equation, one of the things that this equations does,, is that it tells you how omega, frequency and period are related between themselves but also with k and m, so, it's five variables, so, what is the equation going to look like? What happens to f if m doubles? What happens with k if t triples, all this kind of bullshit, so there're five variables here, so you can imagine there's like a crazy amount of combinations, right? One number that's not here is the amplitude, okay? So I want you to write here, amplitude not in big equation, what does this mean? This means that amplitude does not affect t, f or omega, and I meant to make this point earlier but I just solve the equation I remember, this is really important, the chances are there'll be a question like this in your test, is like multiple choice, very simple, is like 80%, this is one of the biggest sor of conceptual point here, why? Because like everything else in physics, anytime something is counterintuitive they put it on the fucking test, right? so, here is the idea, I have a spring and I pull it 2 meters ad I release it and it takes 5 seconds, if I pull it 4 meters it still takes the same amount of time, right? And most reasonable people at first will think, well, you're pulling more so it's going to take longer and come back, right? But the idea is if you pull more, there's more force no it's going to move faster and the fact that you are pulling more it's more space to travel, more distance to travel but it's also moving faster, the two cancel each other out and it ends up taking the same amount of time, okay? So that's kind of almost cool that the only thing that the time depends on is the spring coefficient and the mass, if you're given a bigger mass it changes time, but you can pull however long you want, it doesn't matter, it always take the same amount of time, every single time, cool? So, that's kind of interesting, so, one questions is going to be, you double the amplitude, what happens to time? Nothing. What happens to frequency? Nothing. What happens to omega? Nothing, okay? Amplitude doesn't change these guys, it changes another things but not these guys, okay? And I did this as a first question, alright. So let's keep going here, I now have a vertical spring that oscillates at 2 hertz and I want you guys to solve this one as well, actually let me solve this one, just for the sake of time, this is a vertical spring but guess what? It doesn't matter, vertical springs behave almost exactly like horizontal springs, which means everything we've talked about stills works, and you can basically ignore the fact that it's a vertical spring and not really let that fuck you up, right? It's just a regular, all the other questions still work, that's good stuff, so the mass, I'm even not going to draw, I'm actually just going to do this, because you can think of a vertical spring just the same, vertical spring would be like this, but if you think too much about this you might be like, oh but now there's is an m, g, pointing down, and it's weird, whatever, you know, works exactly, almost exactly the same, so the mass equals 4, what is this? 2 hertz, you have it, you have it, it's a force but you meant another f, frequency, if the mass is moving with 10 when it crosses the equilibrium position, find the system's period, amplitude blah blah blah, so, every time a question gives you a number you need to figure out what does that number mean, right? Because most of the time they give a number you're going to use, there's those deep questions where like they give you a number you don't really need it, right? But most of the time you need it, so, it might be a good idea to think about what is that number or what's the best use I can make for that number, so it says, the mass is moving with ten at equilibrium, so I can say velocity at equilibrium is 10, is there anything special about that velocity? What is it? What's special about velocity? Yeah, good, that's the maximum velocity, so, this is a fucked up way to tell you the maximum velocity, because if you don't know that maximum velocity happens at equilibrium you don't know that that's the v max and now you're fucked because you can't solve this, right? So all the little things kind of connect so this is just telling you that v max is 10, okay?

Now, general problem solving skill in physics, whenever I give you a problem and I'll say this twice, whenever I give you a value for which there's an equation associated with that value, you want to write that equation, so, whenever I give you a value and here I'm giving you v max, when there's an equation associated with it, is there an equation associated with v max? Yeah, what is that equation? V max equals A, w, so, you want to write that out, so if I tell, v max is 10, you write, A, w equals 10, why? Because now that unlock, I can solve for A or w depending on which I have, does that make sense? So, if I told you, for example, the maximum acceleration of whatever is 5, and then you write A max equals 5, and then you write, instead of A it's A equals 5, cool? And then you go from there, does that make sense? You have to figure out what's this number they are giving me, is there anything special about it? And I will place then some values, cool. If you didn't figure all that out hopefully you will get stuck at some point and come back, and be like, what the fuck, let's look at this shit here, okay 10, and the you kind of hustle your way through, because you're not alway like see the light right away, okay. Let's find period, period is big t, again, whenever you're solving these problems I'm just kind of go and write here in this corner because you're almost guaranteed to need this, okay. You start here, if you don't need this then is just, it's going to be a little harder, right? But chances are you are going to need that, and if you write it and you don't need it at least you know you don't need it, right, so you know it's not that, it's everything else, but it's going to work, look at this equation, to find the T I need to know f, w, or k and m, okay? What's sort of the easiest way to get there, do I know k and m? I know m but no k, I know frequency but that's it, we're done, it's actually much simpler, you don't have to think it very much, you know frequency is inverse of inaudible, so, sorry about that, so you know frequency period is the inverse, so it's 1 over 2, 0.5, really silly question, if you knew you shit this would be super easy, inaudible the frequency, you don't have to go around running looking for twelve different equations, what's the amplitude? Amplitude is big A and if you look here, there is no amplitude here, that sucks, so you're going to have to find somewhere else, any ideas of where I might find amplitude from? Right there, right? I gave you v max, it's exactly where going to find the amplitude, if you want A we need w, do we have w? No, but can get get it? Yes, okay? So it's just a fucking little puzzle, you get tusk here, you go there, whatever, there's no way, there's no order to this, just fucking chaos, here's the equation go fucking find some variables, crazy, right? There's no like, do this, do this, do this, so, here I want A, so A is 10 over omega, and that's a really good skill, by the way, to have like math problem solving classes or even for this for the end of semester it's measured is the ability of like you just fucking know all the shit going on and find some equations, plugging in all together, it's going to be really useful in this chapter and next chapter as well, so, w, I can find w, using 2pi f, 2pif is 2 so this is just 4pi, and 4pi is 12.7 or something, 12.6 so, that goes here, meters, okay? So, just like pure hustle, get an equation, plug it in, get stuck, find another one, whatever keep passing around, plus knowing a few things about inaudible, cool. So, you got amplitude is 0.8, maximum acceleration, maximum acceleration is A max, A, w square, I know A, yeah 0.8, by the way guys, don't confuse acceleration a, which is a lowercase a with amplitude, which is uppercase A, right? So I know that's silly but it happens, you'd be surprised, so omega is 12.6 square and this is going to be like 125 or something, 127, meters per second squared, boom, right, so you have to be comfortable, with these random fucking weird problems, okay. So, this is a classic rhine old question, when object of mass m is attache to a spring of force constant k, it oscillate horizontally with a period of T, something like this, k, m and then the period is T, you replace the object wit the new mass or m, and I want to find the new period in terms of T, okay, so, what's going on here? My new mass is 4m ans my new period is box old period, okay? It's one of those problems, remember those? If toggles what happens to that? Blah, blah, blah, the steps, I hope you remember you identify what's changing, what's your target change, what's your giving change, and eventually, ultimately what you want is you want an equation that connects these guys, okay, vast majority of these problems, that's going to be the big daddy equation, okay? So, let's write that, omega is 2pi, f, 2pi over T, square of k over m, this is like 99% of the time, that's the equation we are going to use for this type of problems, inaudible, so obviously this has like four pieces, you know, the equations is supposed to have left equals right, there's multiple pieces here, which means we are going to pick 2 parts of this equation, right? So you want to pick a part with an m and a part with a T, the reason I would write the whole thing is that we see the whole picture together and you should remember that way, so right there, so that's what we're going to use, 2pi over T equals the square of k over m, so that's my equation. Now remember, before I plug it in numbers into this, what do I have to do? You remember? I have to fix this equation, what does that mean? Isolate the T, awesome, right? So I have to leave t by itself before I can plug in numbers, okay? There's two ways you can do this, the sort of traditional way is you move t up, you move this crap down, and then you move some shit around, I don't like that, okay? What I like to do is kind of weird and unusual but I'll show you and you get to the answer quicker, and it's cleaner, and it's totally legal, I can flip this as long as I flip this, okay? So I'm just going to inaudible, because you want the T to be on the top, so T over 2pi, let's put a little flip thing equals k over m, weird because you're used to see, I'm sorry, inaudible, that's really weird, flip, that was like weird because you're used to k over m, and I saw, oh it's 2 k over m, cool? So the two flipped and then if you want T by itself, what you do? Move 2pi over, okay? So T equals 2pi, m over k, by the way, this equation here, in it's final form will be on you equation sheet, it will be there, they'' give you this and this, w equals this, they'll give you a few of them, I just like to put it all together in this formula here, so won't have to derive that, cool. So, now what's happening here? The m is quadrupling so, what do I do? I do this, T equals 2pi, let's put a box in front of the m, and remember, we stick in the number, which number goes where? 4 in front of the m, there's a few ways you can do this from now, I think my preferred method is to just delete everything else, and you just kill everything that's not a box, right? You drop like a bomb in there, the force shield, I don't know, some stupid joke, but the square is still there so what's the answer, that means that you new period is w up here, okay? So, I know this took a little bit but that's because I was explaining very slowly, but basically it's going to tell you if this happens what happens to this? We write an equation, most likely an equation like this, you pick the 2 parts you need, isolate you variable, plug in numbers, inaudible, and the homework has like inaudible, like the first three questions of homework are just shit like this, he likes this a lot, okay. An object of mass m is suspended, suspended implies it's a vertical spring, by a spring that vibrates with frequency f, in other words you got this, and it vibrates with a frequency f, right? Don't think about it, this means too much, it's like that, so is just, you know, you are going to write your, so, now my frequency, when a second object, sorry, when a second object is attached to the first, so, here's a first mass and now a second object is attached to it, mass 1, and now there's a mass 2 here, over here, the system now vibrates a at the frequency of f over 4, okay? So what's the idea? Your new frequency is one quarter of your old frequency, does that make sense? And I want to know what is the mass of the second object in terms of m, s, if the first object was m, what is the mass of the second object in he terms of m? So, is it 3m, 2m, 4m, right? Something like that, you can't, you can't like inaudible, you have to resolve. Now, here is what makes this question hard, this question is really hard, right? Like relative to the other stuff we have done because it's really tricky, what makes this questions hard I'm asking not for the total mass but just for the second mass, remember when I told you earlier, the spring only sees everything, so whenever you solve this and you write you equations for this you are going to get, you have to look at the total mass here and the total mass here, so, what you really are going to do is find this total mass and then you have to figure it out from there, like subtract or whatever, right? Now I don't want to give that away before we get that, but what I need you to understand is that when you have two masses understand that you can only calculate everything together and then figure it out, like divide or subtract or something like that, okay? So, really what you're going to do is if my frequency is now a quarter, what is my new total mass, okay, so, what is my new total mass here, okay? So, I'm going to call this mass total, just to be different from m and then mass total, s, what is my new total mass, let me write this inaudible actually, what

is my new total mass as a, relative to my original mass, is there an equation that relates these two guys? Yes, which one? So, I start here and the I realize that out of here I want an f, it's right there, and I want m that is right here, so, exactly what you said, this, which variable do you need to solve for? The m, okay? I got to get the m out of there, which is a mess because it is at the denominator of the square root, so the first thing I'm going to do is square this thing, so this becomes 4pi square f squared equals k over m, agree? And now let's move stuff around, so look, I move the m up so it's on the left right side and then I move this crap down, so, m is k divided by all of this, is this equation ready to go now? It is because m is by itself, son I'm going to rewrite it here with little spaces and shit, the only number that's changing is f, so you are going to put a box in front of f, remember, the box goes inside of the square because the square will affect that number, what number do I plug in front of f, the f, 1/4, what happens to that number? It gets squared, so 1 over 4 is the same thing, squared is 16, okay? So look, let's kill everyone else, it sounds terrible, so, this is 1 over 16, so far so good? Agree? 1 over 16, and what does that simplify to? 16, right? If you are not sure put that in the calculator, don't do that shit in your head because it's all kinds of, you know, shit that could go wrong, and even when you put this in a calculator you have to be very careful because if you do it wrong you get a different answer, s, you got a 16 here, this means the following, that you new mass, and remember, the spring sees everything at once, this means that the new total mass is 16 times the old total mass, now, in terms of m, the first mass had a mass of m and the second mass has a mass of? Of what? But how much, how many m's are here? This is one m and this whole thing is how many m's? Where you get two? Don't think two mass, so, okay, so, the mass of this, right, m1 is big M, this whole thing combined here is what? How many of these m's are there? In terms of numbers, if this was 1 kilogram this would be 16 m's, okay? Because of here, the new total mass is 16 times the original mass, so if the original mass was m1, then I know that m1 plus m2 must be 16M, does that makes sense? So, what is that m2? It's here, if m1 is M and the m1 plus m2, 16M, then m2 is, yeah, which is? 15M, this is weird because you mass went from 1M to 16M, so what is the additional mass? What's the mass of the second object? It's the difference between the two, so it is 15, that's why this question is hard, because it's not enough that you go through the standard process of figure out, your mass in now 16 times greater but you also have to figure out, the difference is 15 because it went from 1 to 16 so the difference is 15, cool? If this like makes sense like 80% then you're in a really good shape because this is one of the hardest questions you might see, in fact, is kind of like that you won't even see that.

Plot the following oscillation on a position vs. time graph:
x(t) = (1.3 cm) cos((1.2 ms-1)t + 2.3)

A 250 g mass on a spring oscillates with the following equation:
x(t) = (2.8 cm) cos((3.4 s-1)t)
What is the spring constant of this spring?

A mass undergoes a sinusoidal oscillation with a frequency of 50 Hz and an amplitude of 7.8 cm. At a time t = 1 s, the position of the mass is x(t) = 5 cm. Write an equation to describe the positon of the mass as a function of time.

An object executing simple harmonic motion has a maximum speed of 4.3m/s and maximum acceleration of 0.65m/s2. Find the period of this motion.
(A) 0.151 s
(B) 6.6 s
(C) 1.05 s
(D) 42 s
(E) None of these

The position of a mass that is oscillating on a spring is given by x = (17.4 cm) cos[(5.46 s-1)t]. What is the angular frequency for this motion?
[A] 0.183 rad/s
[B] 5.46 rad/s
[C] 17.4 rad/s
[D] 0.869 rad/s

A simple harmonic oscillator has amplitude 0.72 m and period 2.2 sec. What is the maximum acceleration?
A. 0.934689 m/s2
B. 0.327273 m/s2
C. 5.87282 m/s2
D. 0.14876 m/s2
E. 12.9202 m/s2
F. 2.93641 m/s2

A block with mass 5.0 kg moves on a horizontal frictionless surface. The block is attached to a horizontal spring that has force constant 90 N/m. As the block moves in simple harmonic motion, its maximum speed is v = 3.0 m/s. How long does it take the block to move from x = A to x = 0?

A mass is oscillating on a spring with a period of 4.60 s. At t =0 s the mass has zero speed and is at x = 8.30 cm. What is its acceleration at t = 2.50 s? (radians!)
a) 1.33 cm/s2
b) 0.784 cm/s2
c) 11.5 cm/s2
d) 14.9 cm/s2
e) 0

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