Ch 15: Periodic Motion (Oscillations)See all chapters

Sections | |||
---|---|---|---|

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

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

Oscillation Equations | 61 mins | 0 completed | Learn |

Energy in Oscillation | 30 mins | 0 completed | Learn |

Vertical Oscillation | 27 mins | 0 completed | Learn |

Simple Pendulum | 22 mins | 0 completed | Learn |

Concept #1

**Transcript**

But the basic idea of this chapter, there're three things in this chapter, there is the horizontal mass spring system, which is a little spring and there's a block on it and it goes left and right, there's a vertical mass spring system and then there's a pendulum, that's all the chapter has, those three things and typically what we do is spend 90% of the chapter talking about this, right? And that's where you've learned and there's two things that are just sort of additional and they, inaudible very similar to this, cool? The number one thing I need you to remember is the big equation from this chapter, omega is 2pi, f, 2pi over T, square root of k over m, okay? This is the big equation from this chapter and it takes care of almost everything, remember when I solved these problems? I ussually draw a thing like this, negative A, zero and A, and there's a lot of things that you need to remember here, okay? There's about 12 questions in this chapter, inaudible to use which and also the fact that when I pull and release from 1 meter that's the amplitude, period is every little piece of this is a period over 4 and I'll just summarize this, doing the exam, if you just try to refresh you memory so that we can do this. Now, when you a vertical mass spring system, something like this, it behaves almost exactly like a horizontal one and it's actually simpler than you might think, a lot of people think that this is harder because now instead of just having force of the spring you also have m, g pointing down on you but it's actually simpler than what you might think, okay? In fact, sometimes you can solve these problems without even realizing the vertical issue, if you think of this, for example, this equation where it's just the same whether you're horizontal or vertical, so, sometimes the fact that is vertical motion problem, you have a spring on the vertical axis, makes no difference, okay? The only difference or the main difference between these two problems is that amplitude is the additional, and I'll explain what this means in a second, is the additional stretching distance, additional stretching distance, not the hanging distance, I'm sorry? F, s, sorry. Okay, so remember this problem, what was amplitude, amplitude was, the block is here at 0, I pull 1 meter and I release, right? Now, here the amplitude is the additional stretching distance, not the hanging distance, for this to make sense we are going to work out this problem, that's best we can do is just to show you a problem, okay? So, remember that one sentence is different, everything is exactly the same, let's see, here, let me just sort of section this off, we don't have a lot of space here, so we're going to have to be conservative with it, you hang a 0.5 meter spring from the ceiling, so let me draw that here, okay? And I'm going to say that this is 0.5 meters and, cool? So, when you attach a 5 kilogram mass to this spring it stretched by 0.2 meters, so how long is it now? 0.7, right? So, let's draw it a little bit longer and I'm going to say that this is now 0.7 meters, that's because they attach the 5 kilogram here, cool? And it says, next you put an additional 0.3 and release from rest, how long is it now when you release it? 0.10, it's actually 1, right? So it's actually 1, and you're going to release it from 0, from rest, you always release it from rest, by the ways guys, that's always going to be there, release it from rest and I'm putting parenthesis just so you know that's always like that, doesn't mean anything special, okay. So, first thing is, based on that one sentence I gave you up there, which is a key thing to remember, what is the amplitude of the system? Now, I realize that's part C over here, I'm actually going to talk about that first, what is the amplitude of that system, of this system, easy one, 0.3, why is it 0.3? Exactly, it's the additional stretching distance. So, the point here is that is not the, the amplitude is not the 0.2 right here, it;s not the 0.5, it's how much longer you pull this, okay? In this system here this is called the equilibrium, the 0 in the middle where it would be if you don't do anything to it, it just holds the equilibrium, well, this distance, the idea is that you have a mass, the equilibrium position is not here, here you don't have a system, right? You just have the spring by itself, and there would always be like the spring goes back and forth because of your mass, so here the spring just stays there, once you add a mass it's a mss spring system, the problem really starts here, so this is the equilibrium position, okay? Not the 0 but the 0.2 is the equilibrium position, okay? So, the 0 of the problem is actually here, okay? Equilibrium, equilibrium is when the mass hangs naturally, okay? So when I call the natural hanging distance or the hanging distance over here is 0.2, so this hanging distance x2, I'm sorry x1 is just 0.2, the amplitude is my x2, which is 0.3, I call one of them the x1, the other one x2 just for that they're different x's, does that make sense? So x1, sort of in sequence, first you put a mass, it goes over here 0.2, so it's my x1, and then you pull another little bit more and that's your x2, that's 0.3, okay? So, the amplitude is you additional put which is 0.3, not 0.2, not 0.5 just 0.3, but this also means that this system will oscillate, in other words, it's going back and forth around this point here, this is the middle here, okay? So look, it's going to go like this, 0.3, 0, 0.3, does that make sense? So this would go, let's say 3, 3, 3, 3, 3, this is going to go 0, 0.3, 0.3, 0.3, 0.3. So, when it's at its highest point, how long is it? 0.4, yeah, exactly, right? Because look, 0.5, it lowers to 0.7 and now you put 0.3, so, it's going to go 0.7, 1, 0.7, 0.4, there's no fancy equations it's just plus or minus, does that make sense? So, it's going to go amplitude down and then amplitude up of 0.3, which means that would have a length here of 0.4, if you wanted to do this, the math is like 1, plus, I'm sorry, math would be the original length 0.5, plus how much it stretches, 0.2 minus the amplitude 0.3, that's really annoying to think of it as an equation, right? It's just kind do the number in my head, so it's 0.5, 0.7, and then 0.3, 0.3, cool. So that's the general idea, of what's going on here, let's actually solve this problem, there's a few other key things you need to know here. So, here it says, find the spring's force constant, does anybody remember? We already got C out of the way, does anybody remember what force constant is? What variable is that? The k, awesome, so, force constant is k, how do I find k? Sort of, okay, good. So, you might remember at the beginning of this chapter I did a one page summary on force and I really emphasized when you have a vertical spring and then you let that block so that it hanged low, this is at equilibrium, equilibrium means forces cancel, which two forces are cancelling here? M, g is going down and then there is a force going up, what's that? What is it? Right, yeah, good, good, so, it's the force of the spring, and what are the variables for the force of the spring? Force of the spring equals k, x, good, so, I need you to remember that whenever you're at equilibrium m, g, the vertical spring m, g equals k, x, so equilibrium, so m, g equals k, x, and this is you x1 by the way, in this case you have two x's, this is you x1, which is your hanging distance, how long it hangs, right? There's a joke there, sure, inaudible, cool, now I can solve for k. Alright, so please remember this, please remember this, whenever you have vertical spring at equilibrium you can write this equation because the two forces cancel, right? I made I big deal about that at inaudible, okay, cool, so, k equals m, g over x, mass is 5, gravity we're just going to toss just a 10 in here and x1 is, what's x1? 0.2, right? So, notice how amplitude is a number, x1 is a different number they're both useful but for different things okay? This 0.2 need this to be able to figure out what k is and then you need this so that you know like sort of how it's moving exactly, okay? So, this is 250, yes, newton per meters.

Now for part B, there's no good, no, there's no equation or no good equation, the best way to solve some of these questions here is to actually, inaudible thing there, is actually just kind of eyeball what's happening, so, look what it says, it says, what is the spring's maximum deformation, maximum deformation, there's a lot kind of numbers you can use here, is it 0.2, 0.3, 0.5, 0.7, 1? So, what's the definition of deformation? It's change of length, so it's the maximum change of length, what is the maximum length? What's the maximum change in length? Yeah. So, you started with 0.5, that's the original spring right? And then you put another 0.2 and then another 0.3, that's the maximum deformation, does that make sense? And again, there's no equation for this, so this, you can derive an equation but inaudible remember that, right? It's better to just look at it but basically the maximum deformation is x1 plus x2, right? Which is you hanging distance plus your, 0.2 plus 0.3, which is 0.5, or you can just look at it and say well, it started 0.5 and then it went all the way to 1 so that's the inaudible, cool? Part C, we have the answer it was the amplitude, Part D, how long will it take to the mass to reach it's maximum height for the first time after being released, okay? Before I do and I'm really sorry I'm shuffling this but I'm kind of on the fly here inaudible for a better sequence for this questions, I want to do e first, okay? Because e is really easy and then we're going to sort of solve inaudible the rest of the time doing D, inaudible apologize for that all shuffling around inaudible. At its maximum height how far from the ceiling is the block? We already did this, what is that? At its maximum height how far from the ceiling is the block, so the block here is 0.7 away from the ceiling, right? So, 0.5, 0.7 and I go 0.7 plus 0.3 and then 0.7 minus 0.3. So when it's at its maximum height how far are you from the block, I mean, from the ceiling? 0.4 right? So, this is my 0.5 here and then this is 0.4, it's the same number that we talked about that I asked you, whats the highest inaudible, right? So, this is just 0.4, which is the same as asking what's the minimum length of the spring, okay? So, the way we did this is just you are at 0.7, plus or minus 0.3, this is basically what the system is doing, so the minimum is 0.4 and the maximum is 1, okay? I don't want you to inaudible, too much about exactly how these questions are going to be asked, at least not for now, right? We inaudible a test, the exam of you, I just want you to get the basic idea of the fact that x1 is the first number and x2 is the second number and just goes up and down like this, okay? And then you just look at these numbers and then you just sort of imagine this spring moving back and forth and you add some tracking numbers, which? This? Yeah, that's the length of this when you're all the way at the top, right? Which is the same thing as asking what's the distance between the top of the ceiling and the block, right? It's the same thing as the length of the inaudible, okay? So, what's the key idea here, x1 and A are different and you have all these kind of weird questions you might get, let's talk about time quickly, so, how long does it takes to the mass to reach its maximum height for the first time after being released, let's draw one of these things, remember how we used to draw the 0 and the amplitude is the same thing now, 0 and the amplitude, technically this one is positive because it's on top, this negative because is at the bottom, okay? Where did the block start? At which one of these three points does the block start? Like when you release it and it starts moving, where do you release it from? In the negative, right? It hangs to at the equilibrium and then you pull a little lower and you release, okay? So the block starts from here, that's the initial position, and then it goes which way? Up, right? And then it's going to do this and then going sort of back and forth like this, okay? So, the block starts here, this is x initial or I guess y initial because it's in the y axis, and look what the question is asking, how long to reach the maximum height? So, it's asking for maximum height, so, it's asking for the time to go from here to here, right? Agree?Now, I want to remind you of this, I'm sure you will remember how we've done this right after last review, but since it's been a week I want to remind you, every time you get a question in periodic motion, oscillation, spring going back and forth that they ask, how long does something takes, you always answer that in terms of? Inaudible in terms of period, every t question you're going to solve it in terms of period, why? Because there's no direct way to find t, there's a direct way to find big T, okay? So, every time I ask you for t you need to think in terms of big T. Now, question, what fraction or multiple or whatever, how is this time from here to here related to the total period? Over 4, cool. So, over 4 is from the middle to the end, what is it from end to end? Half, right? Good. So, remember period is the time to go from one point back to the same point, okay? So, what you have to find, first thing you need to do is, okay, well, I need these to here, which is half of the period, now let me go get the period and the I divide it by 2, does that makes sense? So, every time in these problems, every time you are asked for time you're always going to be having to find the period first and then divide or multiply, does that makes sense? Cool. So, the t that I want here, delta t is going to be period over 2 but the first thing we need to do I find period, how do I find period? Someone said the letter, w, right? So, w equals 2pi, f, 2pi over T, square root of k over m, this equation works whether you're on a horizontal or a vertical spring, it just always works, except for the pendulum inaudible, okay? So, I'm looking for T, I know k and m, all I have to do is solve for T, okay? Now there's two ways you can do this, to solve for T, one, you can do it my way actually he's going to give you the equation for T, right? So, at that point you just look at his equation, which is ready to plug in all the numbers, okay? But the way I do this, I kind of write only these two guys but I want to solve for T, instead of throwing T to this side and then throwing the root square over here and have this look super fucking messy, I flid these two, okay? So, instead of T, instead of 2pi over t, I do t over 2pi, and you can do that to the left of the equation as long as you do that to the right of the equal sign, which means that instead of square rot of k over m it's going to be m over k, and I'll solve for T, T is just 2pi square root of m over k, which this equation here will be on your equation here, so you can just inaudible, okay? Period of the simple harmonic motion over mass spring system is just this, but even though, again, even though he gives you this equation I still think that's very, very useful to remember this, because it connects all these things together, okay, that's like a inaudible, that's like fifty percent of the chapter in that little white box, yellow box, cool. So, now it's plug in the numbers, T, 2pi, the mass is 5, the k is 250, and let me do this, and I got a period of 0.888, it's like inaudible commercial, and, so the time that I need is 0.444, right? Because time is the period over 2, 0.444. So, whenever you are looking for the time, whatever two points you are given it's always going to be, find period first and the divide by 2 or multiply by 2, something like that, does that make sense, by the way, this part of the problem had nothing to do with vertical springs, this will report exactly the same for anything else, okay? The only thing that was specific to vertical springs was as you realize at this 0.2 is the hanging distance and this 0.3 is what the amplitude inaudible, okay? So, it's an overview of what to expect, now let me give you a question that you might see something like this actually on your homework, I haven't inaudible yet, but you might see some of this in your homework, which is inaudible sounds hard. Any questions for this first part, you guys got a general idea between the differences, cool. A chair of mass 30 on top of the spring oscillates with a period of 2 seconds, so let me draw this, a chair on top of a spring, something like this, okay? Has a mass of 30 kilograms and oscillates with a period of 2 seconds, so, I'm just going to put here period is 2 seconds and then I want to know what is the force constant, that's part A, then it says, you place an object on top of the chair, so the idea is that the chair itself is 30 kilograms, let me just kind of, that looks like there's a box in front of the chair but just imagine this is a chair, it's got like a thick sitting pad there, and now you got an object on top of it and now it has a period of 3 seconds, okay? So, imagine a spring and it's got to, basically you can think of it as one mass of 30 and then another mas of 30 plus something else, right? And if you were to push down it and release it's going to go, right? And it;s going to take 2 seconds to do a cycle, if you add more mass to it, it takes longer, okay? And the idea is that just with that information you're able to find the object's mass, the classic application of this problem is something, is when you are weighting an astronaut and you might see that, we already talked about this, that's how astronauts in space are weighted because their apparent weightlessness so they're floating, they feel no ground, because there's no gravity they are floating whatever, so you can't use a regular like scale or whatever, so the way that the weight themselves is by attaching themselves to some sort of like spring system and calculate it that way whatever, so, you can weight yourself without gravity that way, cool? But let's tackle this first part here, let's find the k for this system, okay? This is much simpler than you think, any ideas on how I can find the k? There are about twelve equations, what is it? Okay, so if this thing is at equilibrium here maybe I can use the fact that m, g going down cancels with k, x, okay? That's a really good idea, I'm glad you thought of that, when you see a vertical spring at equilibrium that's probably the first thing you should think about, the problem here, it's not going to work, but that was good, if you try to do this you realize that you don't know what x is, okay? You don't know x, so that's no going to work, however, what variable you are given? T, and in physics problems if you don't know how to solve something, look at what you're given because it's usually a clue, sso, we're probably going to find an equation that has k in it and then either m or T or both, right? Sometimes there's extra bullshit that you don't need, any ideas? The w equation, right? The big equation. So, this question, to solve this question you can basically just use the big equation and not even think about it and the answer is inaudible, okay? So, again, if you ever not sure of what to do, just look at the equation you'll surprised how often you get the answer right, okay? It ties everything together, notice how there's a period, here is the mass, which I have, and here is the k that I'm looking for, right? The equation for the period if you derive yourself, you flip this into t over 2pi and then you flip this into k over m, you don't have to do that because the equation is in the equation sheet and then you get T equals 2pi, m over k, T equals 2pi, the mass is 30, oh crap, we're actually looking for k, sorry, I got like super excited, sorry about that, I didn't realize that we're actually not looking for, I went through that process but this is not good, we are looking for k, so inaudible, okay? So, sorry about that, how do I get that k out of there, correct, so the first thing that I'll do is square both sides, so, I get 2pi over T squared equals square root of k over m, squared, this is 4pi squared, period is 2 equals k over m, m is 30.

0 of 1 completed

Concept

The natural length of a spring with a force constant of 100 N/m is 50 cm. If a 100 g mass were suspended vertically from the spring, what would the equilibrium length of the spring be? If the springer were stretched 10 cm past its equilibrium length, what would the minimum length of the spring be during oscillations?

A 0.5 kg mass hangs from a 3.5 cm vertical spring with a force constant of 430 N/m. If the spring oscillates vertically with an amplitude of 1 cm, write out the equation of oscillation for the spring, where y is measured from the anchor point of the spring:
y(t) = Acos(ωt) + y0
assuming that at t = 0, the spring is at its longest length during the oscillation.

A spring of stiffness k and relaxed length L stands vertically on a table. A person takes a mass M and very slowly lets the mass down onto the spring. On letting go, he finds that the mass does not move and the spring is compressed by a length d1. The same experiment is now repeated by letting the mass go all of a sudden when the spring is still unstretched. The spring is found to compress by a length d2 when the mass momentarily comes to rest. What is the ratio d 1/d2?
1. Mg / kL
2. 2
3. 4
4. kL / Mg
5. 1/4
6. 1
7. 1/2

Enter your friends' email addresses to invite them:

We invited your friends!

Join **thousands** of students and gain free access to **55 hours** of Physics videos that follow the topics **your textbook** covers.