 Fine so today we are going to start this chapter oscillation and we have how many chapters left I think only oscillations and waves Okay, but these are two chapters that are left for us to cover up and Oscillation will take around two and a half classes and Waves will take about again two and a half classes so in about one month One and a half month your syllabus will get over fine. So right now we are sitting on December of 25th So maybe first week of February your syllabus gets over and second week of February You have your school exams. So you can see how tight the schedule was We have just enough time to complete the curriculum Okay, so if you start a demanding like can we have a longer break 10 days break 15 days break in between Can we go on holidays, you know, then Will not be able to cover up the syllabus or at least then Then I will not be able to talk things in greater detail. So syllabus is so wide that We cannot take a larger break Now if you if you ask me that Why I mean do I like break or not? I'll be the first person to take the break Okay, I teach around nine to ten hours every day So why I will not prefer a break, right? But then it is for you that we are There to help you and be Available to you. So that is why we are we are it is not that we are not giving you break It is that we are not taking the break Okay, it's so easy to take a break. Why don't I prefer that? isn't it and one more thing we will be Will be done with the 11th in maybe first week of February like I said and The class 12th will start maybe mid of March or as soon as your class 11th gets over Okay, so that is what it is and our class 12th syllabus will get over by October that is what the target is So that you have ample amount of time to revise class 11 Then we have to run the crash course also, which we are doing right now with our current class 12th students So crash course is like once we are done with entire 11th and 12th then We will sit with you for every day problem practice three and a half hours every day Monday to Saturday and Sunday will be the mock test for three months. We run that program Okay alright, so Let's begin Has this chapter started in your school already? Has this started already? No Okay, so is there any chapter that is done in the school, but we haven't done it? No, so we are ahead. We are ahead Okay Anyways, so write down this chapter's name oscillation Now when you hear about this word oscillations Honestly tell me what is the first thing that comes in your mind? Have we done kinetic theory or not? Have you finished kinetic theory in symptom? No. Oh Actually, I am teaching multiple batches So I wasn't aware actually in this batch. So kinetic theory will take one more class So just before your school exams syllabus will get over them. Okay, because one more chapter is left So after oscillation will do the kinetic theory. Okay. All right. So oscillations Like most of you said the first thing that comes in your mind is pendulum. Okay Simple pendulum now Oscillation is a kind of movement that repeats after every time Okay. Now if that is the case Let us try to see what are the kind of motions that repeat after some time and what is the Specific about these kind of movement. Okay. So first we will talk about periodic motion Like you said the example of pendulum is a periodic motion also, right? So Can you first define what is a periodic motion? Try defining it all of you What is a periodic motion as per your understanding motion that repeats after Some interval of time that is what someone is saying others a movement that repeats itself a movement that repeats now What what repeats movement repeats means what? What do you mean by motion repeating itself? Are you talking about the Displacement to talk about a velocity or acceleration which part should repeat What should repeat in a periodic motion? Displacement should repeat Velocity or acceleration which one should repeat Okay Now when you are saying it when you are saying it that motion repeats That is actually more accurate than when I ask you which part of the motion is repeating Because when you say motion is repeating it automatically means everything should repeat Everything not just displacement Velocity or acceleration each and every part of the motion should repeat. Okay So write down all All motion variables Motion variables or parameters, whatever you may want to call it Repeat after a Certain period Okay Right now when I say motion variable I mean to say position position should repeat wherever it was it has to come back The velocity should repeat Acceleration should repeat And If there is any other thing like change in acceleration that also should repeat. Okay Now what is this fixed interval of time is called certain in period of interval? What is this called? What is this called? Everyone I just tell me something which repeats in two seconds. Will it repeat in four seconds or not? Something that repeats in two seconds. It would repeat in six seconds. Also ten seconds twelve seconds. Also if it is repeating in two seconds so The the period at which it repeats after itself is not fixed in a way Right. It is multiple of a minimum amount of time after which it repeats. All right, so the The minimum the minimum time for Repeating the motion repeating the motion is Time period Okay, this is the definition of time period and it makes sense to define it for a periodic motion because it repeats after every time interval so Time interval or time period is a very important parameter to look for. Okay. Now tell me a few examples of the periodic motion Pendulum is one of it What is the other example of periodic motion? Any periodic motion you want to talk about anything you're going to school is also in a way a periodic motion Not as of now though Rotation and revolution okay revolution of the heavenly bodies, okay correct so the rotation of earth about its own axis is A periodic motion after 24 hours. It repeats its movement about its own access and After 365 days it repeats it its motion around the Sun Okay, so you'll be surprised to see that When you look around and you see things very Critically, you'll see that most of the movements are periodic only most of it It repeats after fixed interval of time whether you talk about the stars Planet Sun whatever you want to see they repeat itself In fact, we as humans also we have this tendency of repeating the same thing again and again Okay, and I mean that is little philosophical, but if you look at the machines also all the machine they work on the Cyclic movement, which is a periodic movement itself and you have seen that in in thermodynamics chapter the cyclic The cycle what is that it is repeating after event every interval of time Okay, so periodic motion is a very very integral part of our day-to-day life alright, and That is the way that is a reason why we are studying it Separately, otherwise whatever we have learned till now about the kinematics laws of motion work by energy Same thing can be used for the oscillations, but since it is so prominent it warrants a separate Treatment of the movement. All right, but we are in class 11th So we can't learn everything about the periodic motion. So we will talk about a very Specific type of periodic motion, which is oscillations. Okay. Now Can you tell us What is the Definition of an oscillation, I mean whatever comes in your mind, okay You may be right or wrong, but you should not sit quiet. That is the worst thing you can do What is an oscillation? What comes in your mind? Anything random that comes anything swings, okay others Pendulum comes again pendulum comes All right, that is perfectly fine. I mean that is how even in my mind also every time I hear oscillation pendulum comes But in general, can you try defining it? Okay Fine so periodic the oscillation is a periodic motion only but a special kind of it a periodic motion Which is basically to and fro to and fro movement about a fixed point All right about a fixed point to and through motion. What does it mean? Just take an example of pendulum also. What does pendulum do it goes around in this path, right? Pendulum Bob is here right now. Then it is there right now can you identify a fixed point for this pendulum Where is a fixed point? This is position a position B for the pendulum middle of it over here This is the fixed point about which what happens it goes away then comes back it Goes away and then comes back. So this is your fixed point for the pendulum All right, this is the fixed point Back and forth motion For the pendulum like this. All right now fixed point is a very important thing when we talk about the Oscillations and remember when we talk about the Revolution of Earth around the Sun there is no fixed point. So it is a periodic motion only but it is not an Oscillation, okay? This fixed point is the characteristic of all the oscillation So it's very important. So there's a name to it. This fixed point is called mean position This fixed point is called a mean position. Why it is called mean position. What does it mean? Why the word mean any guesses why the word mean? Mean is like the average. Yes mean as in average So basically whatever happens whatever happens on one side of it same thing happens on the other side So it makes sense because this symmetrically placed always it will be symmetrically placed It'll never happen that this time this side something else happens and this side something else happens about a mean position Whatever happens on this side same thing will happen on that side Exactly same Okay, I mean I'm talking about an ideal scenario wherein there is no friction There is no air drag and all those things are absent only the pendulum and Bob is there We'll keep on doing this till infinity. Okay now There can be many types of oscillations as well. All right, the simplest kind of oscillation the simplest type of oscillation is simple harmonic motion Oscillation another name is harmonic motion So the simplest of this is simple harmonic motion, which is There in our curriculum the other kind of oscillations are not there Okay, so in this chapter we are going to study about this simple harmonic motion only okay, in fact Most of the books have don't have a chapter name oscillation. They will write simple harmonic motion or SHM so don't get confused SHM or simple harmonic motion is same as your textbook chapter of the oscillations. Okay Now, let us see how the chapter is arranged then we can probably Again go back to the introduction of the chapter So there are three parts of the chapter part number one part number one is the kinematics kinematics of simple harmonic motion Okay now One thing you need to understand that your Foundation of physics when we started the first few chapters were about the foundation of physics Where in we discuss about what is displacement? What is velocity? What is acceleration? What is force? What is energy? What is kinetic energy and things like that? We got a basic tools. It's like we got a hammer. We got spanner. We got screwdriver in first few chapters Okay. Now going forward The definition of what is force will not change definition of what is energy won't change Whatever basics you have learned in first four or five chapters You're going to use it again and again for different different kinds of situations All right, and you will be surprised to see that it will happen in class 12th also Okay, so here is a chapter in which What is happening is that there is a special kind of movement which is simple harmonic movement For that movement. We are applying our knowledge of kinematics Which we already know our knowledge of laws of motion, which we already know and our knowledge of work energy power also for this particular situation and Similarly, you will see in class 12th. You will study electrostatics So you will again apply your knowledge of laws of motion on electrostatics Your knowledge of kinematics on electrostatics and your knowledge of work by energy on electrostatics So things will become repetitive from here So the first four or five chapters are the most crucial chapters of your physics for an entire 11th and 12th combined after that things are repetitive So part one like I said talks about kinematics of SHM. Now why we are having kinematics of SHM because We will discuss in greater detail later on about the kinematics, but kinematics is what studying a Movement without getting into cause of the movement. So I don't care about how the SHM has been created I have been given position time and velocity. I may need to find acceleration or displacement after certain time So that is what the kinematics of SHM is about. Okay part two Part two of the chapter is The major part of the chapter that talks about the cause Causes of SHM. What causes a particular SHM? What kind of force creates an SHM? Okay, and What are the energy involved in the SHM? Fine, so we'll be talking about potential energy kinetic energy and you can use work energy theorem here Also, your basics won't change force will be still equal to mass and acceleration Your work energy theorem still be same just that probably you will write potential energy in a different way Okay, similarly force also It may be a different kind of force, but force is equal to mass and acceleration still valid This part Constitutes about 80% of your chapter Okay But many times I have seen that questions come from part one Because students think that if I do part two well, my 80% is taken care of 80% of syllabus gets taken care of but if I am the examiner I have to set questions what I'll do I'll think which topics students ignore and I'll frame a question on that topic Because my aim is to eliminate my aim is not to select Because there are many students right? So first I'll eliminate and then select All right part three part three of the chapter it talks about Non-ideal scenarios Non-ideal Scenarios or situations So we never took care of the fact that there'll be damping in the oscillation after certain time The oscillation will die down and it'll stop So in part one part two, we don't care about it in part three. We touch little bit about how these damping will come into effect and also about the forced oscillations, okay, so Here we are talking about more realistic scenarios So just to give you One example when when the earthquake comes You might have seen in the televisions that some buildings Collapse easily some buildings don't even You know get damaged at all The reason is because the natural frequency of oscillation for that building matches with the oscillation frequency of the earthquake Because of that it starts fluctuating violently and that is why it collapsed and most of the time the the tallest building doesn't get affected because that The natural frequency depends on the shape and size of a building when it is tall then We it is seen that the frequency of the tall building doesn't match with the frequency of oscillation of the earthquake Okay, so we're going to talk about these things briefly. Okay, but again We'll start from the Basics first that is part number one kinematics of SHM. Okay Now when we talk about the kinematics of SHM We do not have any Equation right now to start with we don't know anything. All right, so we need to first at least get one equation Okay, it can be Equation for force it can be equation for velocity or acceleration Right some equation to start or begin with we cannot keep on talking about Oscillation philosophically that oscillation should be like this. There are a lot of examples this and that all that is Good to hear but not useful. So mathematically. Let us see how we can Try to model it. Okay, as in till now you can see that till now Whatever equations we have in physics. We have seen the observation and based on the our Observation we have come up with the equation like the way we have formulated The force is going to mass and acceleration We have seen that if there is an effort on what all things it should depend upon and that is the reason why force is able to mass and acceleration Okay similar kind of analysis Let us do for the simple harmonic motion or any oscillation and see Which mathematical relation fits for that situation? Okay, so let us first observe what is going on We'll observe it critically critically means we'll be seeing velocity displacement acceleration and see Is there any relation? Okay, let's see this. Let's assume that this is the path of the Oscillation oscillation is happening about this path This mean position is exactly at the center This is oh All right, and there are extreme positions. Let's say P and Q are the points till where the object goes and then comes back back and forth motion, okay to and fro So can I say that object has to stop at PMQ? object should While it is going towards Q like this. It has to stop and Q stop at Q and then come back Is that true that object should stop? at P and Q All of you agree or not? Till the SHM is going on object must stop at Q and reverse its direction of movement Similarly object must stop at P and reverse its movement. All of you agree with this? Any confusion? Type it So object should stop at P and Q now if object stops at P and Q the Although mean position is an Important parameter P and Q also becomes important. So we'll name that as well So P and Q. They are called extreme positions P and Q are called extreme position. Oh is the mean position and OP Distance is equal to OQ distance and this is called amplitude Amplitude of the oscillation a Okay, now Tell me one thing that if I go like this Turn back and reach there Is it repeating the movement or not? Is it one complete oscillation or not? What is the issue with this? I? Have reached here itself. It isn't it repeating my movement. Is it not repeating my movement? It should go to P also why? My definition of the periodic motion is that the movement should be repeated. I start from O and then came back One complete oscillation. What is the issue with this? Periodic motion. I'm talking about what is the issue with this? Why it is not one complete oscillation everyone symmetry Symmetry doesn't matter. Okay. You should go by the definition. Is it a periodic motion or not? Is it a periodic motion or not? Not talking about oscillations here. Don't get into to and fro. I'm talking about is it one period or not? One time period of oscillation. I start from over and then comes back there. Is it not completing one? cycle all of you type in are all the motion variables repeating itself when I go back at O all the movement all The motion variables Displacement velocity Acceleration they repeat itself or not is velocity repeating itself When I start from O, I am going this way when I'm coming back. I'm going that way is Velocity repeating not getting it what I'm asking. Okay. I should take your names then Ananya What is the issue? Anusha Dave Anusha Dave and What name I've taken Ananya Dave Bantia Sometimes I wonder whether you guys are actually paying attention. Some of you might be eating. I will not be surprised There's a class going on you should have a complete attention No, nobody's going to repeat it. Okay, and you may think that you have a recording You can watch the recording later on Ask yourself how many times you have watched the recording? Very honestly, okay recordings doesn't help okay it is just to See things if you have a doubt at a particular lecture then go back and see that thing again But if you miss the live thing then recording won't help you can see for past one year You had recording how many times you got benefited out of it? Okay, so the Huh, so velocity doesn't repeat directional velocity is changed. How can it be repeating? So it has to go there. This is first movement Then it has to come back Now everything is repeating except velocity. It is not a one-time period Goes like this goes like that. So it has to go back to P like this and Now when it comes back, you will see that even velocity is repeating Okay, now when it comes back Even velocity is repeating. So in total there are four distinct movement Okay, these four movement will repeat in every cycle Oh to Q Q to O O to P and P to O Fine. So let us try to see Little bit more about the movement of it. So we'll draw a table like this all of you draw it with me Four columns it should have draw it like this This is over to Q Oh to Q. This is Q to O This is O to P and P to O Okay, all we have to do is simply see the directions of the Position Velocity and acceleration only okay now before we do that exercise We are assuming this direction to be positive and that direction to be negative keep that in mind, okay, and I am tracing the Position not the displacement There's a difference. Okay. So this is you can assume that this is your x-axis X-axis O is the origin of the x-axis when I say position. I am tracing the x coordinate From the mean position if mean position is zero. I am tracing x coordinate of where the object is Okay. So all of you We are talking about x V and a So write down x V a for all of it x V a x V a Now for O to Q For O to Q. Can you tell me x is positive or negative? x coordinate positive or negative all of you should participate When I'm teaching very very basics, all of you participate. Otherwise soon things will become advanced or difficult Then you'll be like, yeah, I didn't pay attention. So I'm not getting it. So don't be in that. Sorry state Participate right now itself when things are easy X will be Greater than zero All right now What about velocity? Velocity direction positive or negative Positive it is also on the right-hand side greater than zero Okay Now talking about acceleration Should it get decelerated or accelerated when you go from O to Q? Everyone should it get decelerated or accelerated? It has to stop at Q It should get decelerated. So acceleration is an opposite direction of the velocity If velocity is positive Acceleration should be negative Now Q to O. We are coming back. What about X? X coordinate Q to O X coordinate How can it be negative O to Q all the X coordinates are positive? O to Q all the X coordinates are positive wherever you go Doesn't matter how you're moving 1 2 3 4 5 all are positive Okay, so Q to O the X coordinate is positive only What about velocity? Velocity is less than zero Okay, now acceleration is in the direction of velocity or in opposite direction Same direction of velocity or not it is Having zero velocity here. It is it has stopped. So does it need acceleration? To move this way it needs an acceleration, right? So a is also less than zero Because a is also that direction and that direction is negative Okay, I think some of you might have learned that deceleration is always negative and acceleration is always positive That is actually not true Deceleration only means that acceleration is in opposite direction of velocity Acceleration only means that acceleration and velocity directions are same if velocity is negative Acceleration also negative. It means that object is accelerated. Okay All right O to P. What about X? Negative or positive X is less than zero. What about velocity? What about velocity O to P? negative acceleration Acceleration is the same direction of velocity or in opposite direction of it in the opposite direction So it has to get decelerated Okay, because at point P it has to stop if velocity is this way expression should be that way for it to stop Now P to O the final thingy X is positive or negative less than zero velocity What about a velocity? positive greater than zero acceleration It is accelerated or decelerated P to O Accelerated so well acceleration is also greater than zero Okay So this is our small analysis of How the things are in the case of the oscillation now I want you to See these relation and find out Is there a pattern a? Pattern which is true for entire movement. Is there a pattern? O to Q Q to O O to P and P to O throughout it There is a pattern which you need to identify. It's like a puzzle Look at these values and tell me. Is there a pattern? Okay All of you can see that Ha good good to see that most of you identified the pattern that X and a they're always in the opposite direction Yes or no? all of you right so science of X and a opposite Now this is true for any kind of oscillation or only for this simple harmonic motion This is true for all the oscillations or only for the simple harmonic motion Everyone it is true for all the oscillations. Okay Now if a and X are of opposite sign Can we say that V and X have same sign while moving away from the mean position? Yeah, you can say that but that is I mean that's not a pattern right because every time There's a bigger pattern The bigger pattern is this X and a are of opposite sign that is so true But when you're telling that V and X have same sign While moving away from it that is that is becoming more specific to one kind of movement inside the SHM It is not a pattern. I will not say that It's a pattern. Okay. Okay, so The bigger pattern is X and a are of opposite sign. So can I say that a is proportional to negative times X to the power n Does it make sense? will this Take care of the fact that a and X are of opposite sign There's a minus sign it takes care of that fact. Okay, so if I say a is equal to negative of some constant times X to the power n Okay, and belongs to integer this can take care of the fact that a and X are always Opposite but there is an issue. What is that problem? What is the problem with this definition? Tell me There is a situation in which a and X need not be in the same directs and need not be of opposite sign What is that? When the body is in mean position Zero has no sign. So a is zero and X is zero. Any other? When X is negative, okay, correct correct so all of you if If X is less than zero if X is negative and and n is even integer Then what will happen to X to the power n? Will it be positive or negative if n is even and X is negative It'll be greater than zero So if this is greater than zero Then what about acceleration? Will it be less than or greater than zero when you use this? It will less than zero. So you can see that X is less than zero and a is also less than zero That is not desirable. Okay n Should not belong to an even integer. Okay, so that's the reason why a is equal to minus some Constant X to the power n where n belongs to odd integers only Then only this is valid Okay Now this is the equation for the oscillation For any oscillation. All right, and for simple harmonic motion Can you guess what is the value of n? simplest n is equal to what smallest or integer is what? one That you can answer one, all right, so we have as such an equation as a is equal to minus c times X Now one more thing Cance this constant of proportionality. Can it be negative? Can this see the negative? No C has to be greater than zero So if it if it is always greater than zero, I will be Double careful how to be double careful you write your constant as Square of something omega square X Okay, so this becomes your equation of the shm or shm condition Accession should be equal to minus a constant square times X Got it. Okay. Now. Tell me What are the locations in which? Exploitation is maximum a is maximum magnitude where a is max What should be the value of X for which a is maximum X should be maximum and what is the maximum value of X? amplitude right a is maximum at extreme positions Okay, that is what a is you can see that Velocity is zero at extreme position, but a is maximum and Where a is minimum or a is zero? a is zero at what at mean position where X is zero What about velocity is velocity maximum there is velocity maximum look at the movement Till from oh till it was coming to oh it was accelerated as soon as it goes this way it starts getting decelerated Okay, so It's velocity is maximum at oh from q while it is coming to oh it was accelerated And as soon as it go away from oh it starts decelerated Okay, so that's the reason why the velocity is Maximum at mean position and this is true not only for shm for every oscillation. This is true Okay, so remember these Few things So from here onwards your chapter from ncrt starts, okay? So a is equal to minus omega square X is the equation from which The using which you can say the shm is There now if you talk about kinematics of shm, do you remember equations of motion? What are they? What are the equations of motion? V is equal to u plus 80 s equal to ut plus half 80 square and V square equal to u square plus 2 as Okay, can I use them here? Can I use these equations for shn? Everyone if not why Why is the question only when you say no if yes, then you don't need to Tell the reason because that is not correct We cannot use this the reason is that acceleration is not constant. These are valid only for the constant acceleration Okay, so Don't use these don't Use them for shm Don't even think about it Okay Now what are the options we have? Do we have any other thing that you can use any other thing that we can use other than equation of motion? Can I use the Differential form which is what? V is equal to DX by dt a Is equal to dv by dt? A is equal to V dv by dx All of these equations are they valid for shm or not? They are valid for shm or not these everyone they are valid for Any situation? I mean, this is the definition of velocity and acceleration All right, you can't say that no no a velocity is not rate of change of Displacement you can't say acceleration is not a rate of change of velocity because if that is not true Nothing is true in physics. That is a definition of it now if this is valid for any situation Why we had equation of shm? I Could have used this for constant iteration case also then why do I need that? What is the reason? Why do why I needed equations of shm? So equations of the motion? Why did I use them if I had calculus with me? Why we learned these I Could have used calculus to solve everything What's the reason no one knows easier to use As simple as that Why will I break my head to again and again use calculus integral if efficient is constant? I can directly use it Okay, it's like if It's like, you know, if you If I give you let's say If I give you wooden you can make Table and chair out of it If you have ready made table and chair out of it, why do you want to make? Table and chair from the wood daily and then use it Okay, you can directly use the wooden table and chair which is already made for your purpose So that's the reason why you know constant acceleration is a very common scenario So for that scenario common scenario, we have a set of equation You don't need to look at the calculus if expression is constant You can directly use it. It is easier to use it and We have to teach you in class 9th also same thing So we cannot introduce calculus to you in right in class 9th itself So that's why it makes sense to create these equations of motion Now even for shm Okay, for shm also shm is also a common type of Motion Although addition is not constant. So what we will do we will We will use calculus to derive equation of motion for shm So that we can use them directly So let's try to see how we can derive it and then we will have some questions based on that. All right So the first equation is already known to you a is equal to minus of omega square x This is equation of motion itself a relation between acceleration and position x Okay, this is the equation of a shm first one How will you derive a relation between velocity and x any guesses How you go about it? We can write V as sorry. We can write a as V dV by dx Acha please take x is equal to plus minus a For amplitude, sorry for the extreme positions Okay extreme position a is the amplitude. So can you try deriving it? Try deriving it yourself get the answer. It's a simple derivation Okay, others done everyone When you integrate you cannot integrate indefinitely Otherwise, there will be a constant of the integral also come in when you integrate Okay, there should be limits upper limit lower limit So V dV is equal to minus of omega square x dx Okay So when you integrate this You put the limits All right. So what is the value of V when x is equal to a when x is a what is V? V is 0. It was at rest. Let's say at x equal to x velocity is V All right So when you integrate it it becomes V square by 2 0 to V This is equal to minus of omega square x square by 2 A to x So put the limit it become V square equal to minus of omega square X is square minus a square All right So V is equal to omega under root of a square minus x is choir Okay, so this is the second equation of motion first equation of motion is this Second one is this Second one is a relation between V and x. Okay. Now you can see that the way it is different from your previous equations of motion these three is Look at it. Every equation has three variables V a and T s t a V a s And a is common in everywhere All right, so you should know what is a but over here you have only two variables a and x V and x so these equation of motion they are better than your previous Equations of motion. All right, although it looks let's say Different because you're doing it for the first time But if you ask me which one is better or easier to use it is these ones Variables are less Okay, now tell me what is the maximum velocity? Maximum amount of velocity is how much? Look at this equation number two and tell me V max. How much it comes in terms of omega and a V max is omega into a It happens at x equal to zero Okay, and it can be on both sides plus or minus while going this way plus going that way is negative So both directions So the way the shm happens if you look at it This is mean position the object goes like this and it turns Goes like this and then comes back like that So everywhere you go, let's talk about this position While going forward it is this way the velocity while coming back. It is this way Here also while going this way. It is this direction while going that way It is this direction. So at every point at every point the velocity Can be in both directions. It depends on whether you're going Towards the mean position or towards the extreme position So you get two directions of velocity plus minus every x only at extreme position Because velocity is zero you have only one velocity, which is zero because it is at rest Everywhere else if you tell me find out the velocity at x equal to two centimeter and Amplitude is three centimeter then I tell you two answers one is Positive other one is negative. Okay, so remember that also So you have two equations of motion a is equal to minus omega square x and v is equal to this Now we will bring in the time factor. We haven't yet Brought the t in the equation. So how will you do that? How will you bring t in the equation? Any guesses? correct So you write v as dx by dt is equal to omega root over a Square minus x square Okay, and then you solve this differential equation How will you do that dx divided by root over a square minus x square is equal to omega dt Okay, by the way, do you know how to integrate this? Do you know the formula for integral for this? What is integral of dx by a square minus x square? If you don't know it is all fine Just a formula. There is no analysis as such Do you know how to integrate this? Okay, no one knows it This integral is actually signed inverse of x by a It's a direct formula Limits will be from zero to t what should be the value of x when t is equal to zero What should be the value of x when t is equal to zero? Everyone t equal to zero. What is the value of x? Some of you are saying zero Some of you are saying zero At t equal to zero. Is it required that it should start from here only? Is it required that should start and go like this? Everyone it is not required. Okay, so it can be anywhere It can be anywhere need not be at x equal to zero So we'll say at x equal to x naught t equal to zero and this is x They will do omega t Okay, it can as well start from here Go like this and like that it travels Okay, so it can start from anywhere. Don't assume that it will always start from the mean position t equal to zero is the observer's description. I can start my stopwatch at Any point in time and say that that is equal to t Zero, okay Now when you put the upper limit and lower limit, it will be sine inverse X by a minus of sine inverse x naught by a is Equal to omega t Okay This is the Have you seen sine inverse before have you have you seen the inverse technometric function? Sine inverse half. Let's say what it is What is sign inverse half? Simple 30 degrees or pi by six. Okay, so sine inverse of anything is an angle Okay, so we will say that let's say sine inverse of x naught by a is Phi some angle phi which is the initial phase Okay, so sine inverse of This is x by capital a is Equal to omega t plus Phi So x by a is Equal to sine of Omega t plus Phi So we will get x equal to a Sine of Omega t plus Phi This is the third equation of motion Third equation of motion is this right and many times in your Questions, they will talk about write down the equation of SHM. All right When you say when they say find out the equation of SHM, this is the equation of SHM. Okay The relation between x and t For some reason, this is called the equation of SHM Now we have got what a as a function of x v as a function of x x as a function of time Can we find out v as a function of time everyone? What is v as a function of time? Can we find out that? It's very easy differentiate with respect to time don't hesitate It's direct v is equal to dx by dt This is equal to what a omega cos of Omega t plus Phi Okay, so we have velocity as a function of time now All right Can you get acceleration as a function of time? Just differentiate it a is equal to minus of a omega square sign of Omega t plus Phi This is your a as a function of time So you can see that a sign of this is x So it it matches pretty well with a is equal to minus of omega square x All right So instead of three question of motion you have five to use So now we'll write it again all of them together everyone a is equal to minus omega square x V is equal to omega root over a Square minus x square x is equal to a sign of Omega t plus Phi Then V is equal to a omega Cos of omega t plus Phi and a is equal to minus of a omega square sign of omega t plus Phi So you have five equations equations of motion So you don't need the other three light So you can see all these equations. They are standard ones Just that when time comes T factor comes in there is this Initial phase phi that is also part of the equation But when we write x as a function of a and V as a function of x It does not matter from where the particle has started Okay, but when you write with respect to T You have to find out what is the value of Phi now talking about Phi a little bit So that it doesn't confuse you when you solve numericals so sign in a worse of x naught by a is Phi Now suppose if x naught is equal to a by two What is the value of five? everyone If my ascension starts from x equal to a by two from here Okay, so you're saying that sign universe of half it will be and this should be equal to Pi by three sorry not pi by three pi by six Now There is a mistake you guys are doing the object can go this way From there and it might be that it is coming back from there Are these two situations same? Will the value of Phi will be same for both? No, it is not same So there are two values of Phi for every position of x Apart from the extreme positions. They're two values of Phi So Pi by six and it can be Pi minus Pi by six right sign of Pi minus theta is also sin theta only so it can be Pi by six or Five Pi by six It can be both When it is going to this direction It is Pi by six when it is coming back. It is five Pi by six Now I can simplify it further to you the angle over here is Zero if it goes this way Then when it reaches the extreme position what the angle becomes Phi will become what? when x equal to a Phi will become Pi by two Pi by two and when it reaches here again, but in opposite direction it is Pi So basically you need to think like this that as I go in this direction The angle should be between zero and Pi by two when I come back the angle starts increasing to Pi and then when I reach here sin inverse of minus This is minus a x equal to minus a sin inverse of minus one You have to write and this will be three Pi by two This angle is three Pi by two when I reach there and when I come back It becomes two Pi over here and two Pi and zero are same Okay, so again, this is zero angle. This is Pi by two angle I come back it become Pi the again three Pi by two and then two Pi So the angle of Phi Goes from zero to two Pi in entire cycle. So you have to see where the object is and Which direction it is moving? Is it clear to everyone type in Just go through it once and let me know if it is clear spend one or two minutes Will extreme position have only one value of five? Yes, because the velocity is zero there It's a unique value Every other place there are two directions of velocity Depending on which direction the velocity is the Phi will be that Slowly and slowly when you solve questions, it'll be you know, it is something which is very straightforward You will get to know that Or if you are a little confused with this kind of thing What you can do is that you can check both x and v for the value of Phi V for that value of Phi at equal to zero the velocity should be negative if it is coming back So you can substitute the value of Phi and check whether the velocity is negative or positive If it is positive, it means it is going this way if for that value of Phi if V is negative It means velocity should be that way Fine, so you can use both X and V also. There is no one method of doing same thing It can be done in multiple different ways Okay, shall I move forward everyone so next thing We are done with the equations of the SHM now we are going to use equations of SHM We will be doing some analysis some numerical on it. All right The first and most important thing that we have discussed in the first slide was time period Which is what we need to find out Time period of SHM Okay, so if X is equal to a sign omega t plus Phi using this equation Can you find out the time period of SHM? All of you the hint is X should repeat for every value of time Okay, this is the graph of X and T All right, so time period is what I'll give you a hint So suppose time period is capital T. So can I write it like this omega t plus Phi? This is equal to a sign of Omega t plus capital T plus Phi. Can I write it like this? everyone after time capital T it will get repeated so I'll add capital T to small t then same thing should happen again All of you agree so I can write it as a sign of Omega t plus Phi Plus Omega into capital T Now I can equate it to a sign Omega t plus Phi if I equate it I'll get t equal to 0 But I can as well add 2 pi to it because 2 pi is the Minimum cycle of the sign So even if I add 2 pi, they should be equal So for now if I compare it, I'll get omega into time period equals to 2 pi or the time period of SHM Is 2 pi by omega So you can see how nicely it fits in Omega square was a constant of proportionality in SHM equation and the time period is 2 pi by omega So frequency is 1 by time period that is omega by Phi Is it all clear to everyone? Everyone please type in I've seen sometimes students say that Omega is the angular velocity Okay, omega is one of the symbols for the angular velocity But in the SHM, omega is not angular velocity Omega is the name of that is angular frequency Don't use angular velocity for omega 2 pi part once again See Although these two are equal But if I equate it, I'll get capital T to be 0 Okay, so what is the minimum angle I should add here so that I can equate them You can add 2 pi angle without affecting any value of this I'm not changing T, I'm adding 2 pi Because it is sign function, it will repeat after 2 pi Adding 2 pi and then equating it So this and that, they should be equal I could have added 4 pi 4 pi is equal to omega into T, but that is not the minimum time When I add minimum angle here for which both are equal Then only I'll get minimum T, so that is my time period Okay, fine, so This is the time period of SHM Now can you do one more thing I draw the graph between acceleration versus X For an SHM, you know that A is equal to minus of omega square X So I want you to draw a graph between A and X This is A and that is X Try doing it Then everyone, what kind of graph you'll get Constant negative slope passing through the origin Passing through the origin, okay So are you getting this? This line, this is not correct There's a difference between physics and mathematics Why this is not correct? Because there's a limit up to which X can go X cannot go beyond A And minus A So you'll not get line, rather you get a line segment Okay X equal to A X equal to minus A A is, this is omega square X, omega square A This is minus of omega square A This is what the graph is Okay, straightforward equation of a line it is Y is equal to minus of Mx You have done it, I think, equation of a straight line Now, I want you to draw velocity versus X V is equal to, I'll write the equation here V is equal to Omega under root of A square minus X square Try doing it, you have to be little analytical here Don't blindly follow the mathematics of it Just visualize how it should be and draw the graph Approximate graph is fine, good enough The hint is, first you draw the critical points velocity Critical points are mean position, extreme positions You have two extreme positions So at X equal to minus A And X equal to plus A The velocity will be how much? Zero So, these two points will lie on the graph or not Velocity should be zero, so I already know two points on the graph Plus A and minus A Do I know any other point on the graph? Everyone, at mean position, what is the velocity? What is the velocity at mean position? Mean position is X equal to zero At mean position, do you have only one velocity or you have two velocities? This is Omega A and you have minus Omega A as well Two velocities Okay Now, for every value of X, let's say if I draw a line like this Will I get two different velocities or not? For every value of X, one positive, one negative Everyone, type in I get one positive, one negative One positive, one negative For every value of X, you can see here One velocity while going this way, one velocity while coming back at the same position All of you type in, is it something you are understanding? So if you join all the points, you are going to get a curve which is closed Going to get a curve Okay, I may not be able to draw a symmetrical curve, but it is a symmetrical curve Last attempt Nice So you will get an ellipse Harsha, why you joined after two hours? Harsha, message So you will get an ellipse, okay? All of you just go through it and let me know, is it clear to everyone? Just read through it once again Harsha, your message, why you joined after two hours? What do you mean over slept? Is it the time to sleep? 11 am Your school starts at 8.30 Okay, is it all clear? Fine, we will now move little forward Next question Is this You have to find the location at which velocity is half of its velocity at mean position Okay, amplitude is given as A In terms of A, you find out at what location the velocity is half of its velocity at the mean position Okay, we have both got something, others V is equal to omega root over A square minus X square Velocity at the mean position is how much? At mean position, the velocity is at X equal to 0, omega into A You have to find out X when velocity is omega A by 2, that is half of the mean velocity Omega A by 2 should be equal to omega root over A square minus X square This is what the equation is Omega gone, it will be A square by 4 equals to A square minus X square Now, you might be aware that for the velocity, same velocity can be this side, can be this side, this way and that way So, you can have two different values of X for same velocity because it is symmetrical about the mean position So, before even solving the problem, you should be aware that you will get two values of X plus and minus Okay, then you should start solving, it should not be that after solving, you are wondering, okay, will I get two values So, X square is equal to 3A square by 4 So, X is equal to plus minus root 3 by 2 times A This is the answer Okay, let's proceed Next question is this You have, if you have any doubt, you can stop me, message immediately, I will answer it This is O, this is plus A, this is minus A for the SHM, okay There is a particle which is over here Okay, at t equal to 0, the particle is at X equal to A by 2 and it is going towards O Like this, at t equal to 0 You need to basically write down equation of SHM So, what I mean by that, when you write X equal to A sin omega t plus phi I want you to find out what is the value of phi for this situation Initial phase is how much Repeat the question At t equal to 0, it is here, it starts going this way, you have to find the value of phi I mean, I think I have already told you that, sometime back, do it yourself the other way The velocity way, not the way in which I have told you Try getting the value of phi by using the other method So, we know that at t equal to 0, the value of X should be A by 2 And at t equal to 0, the velocity should be less than 0 Negative velocity it should have So, when you put at t equal to 0 for equation number 1 We will get A by 2 to be equal to A sin of phi So, sin of phi being equal to half Tells us that phi is either equal to pi by 6 or 5 pi by 6 Phi should be between 0 and 2 pi, it cannot go beyond 2 pi, remember that Then at t equal to 0, if I write equation number 2, velocity is A omega cos of phi Now, I have two options, this and that for X equal to A by 2 Which one will give me velocity negative? 5 pi by 6 will give me negative So, the answer which satisfies both the conditions which are required Is the value of phi to be equal to 5 pi by 6 I think this method is You will be more in control if you use this method Although little lengthy, but you are in more control Getting it, is everyone clear about this question? Alright, so we will take one more So, you have X is equal to 5 sin, I am reading the question only 5 sin pi t plus pi by 3 This is one of the equation of motion Alright, you have to tell me time period of the oscillation The maximum velocity over here And you have to tell me velocity when t is equal to 1 second Velocity at t equal to 1 second is how much, do it Okay, we have got something Time period all of you got it, t is equal to what? What is the formula for time period, 2 pi by omega? How much is omega for this SHM? Everyone, what is omega for this one? Omega is pi You can correlate this equation A sin of omega t plus pi When you correlate, you will get omega as pi So, time period is 2 pi by pi, 2 seconds Okay, maximum velocity, what is the formula? Omega into A, omega is pi, what is A over here? How much is A? A is 5, so 5 pi meter per second is maximum velocity Now at t equal to 1 second, what is the velocity? How will you do that? How will you find this one? Yes, you can say plus minus, why not? But I am assuming we are interested in the magnitude of the velocity rather than direction of it So, what I will do is that, I will differentiate v is equal to dx by dt Alright, so I will get 5 pi cos of pi t plus pi by 3 At t equal to 0, the velocity is 5 pi, not t equal to t equal to 1 cos of pi plus pi by 3 What is cos of pi plus theta? Cos of pi plus theta is what? This is technometric identity minus cos theta So, this will be minus of 5 pi cos of pi by 3 What is cos of pi by 3? How much it is? So, minus of 5 pi by 2 meter per second Minus means it is going on the left hand side Okay Is all of you clear? Alright, so this is clear One last question before the break I don't want to take questions on kinematics after the break We will start something new So, here it is, there is a particle that starts shm At, it starts from x equal to minus of root 3A by 2 So, this is the location at t equal to 0 And it starts moving towards extreme position It starts from this position and goes to the extreme That is the situation Alright Time period of oscillation is given as capital T Time period is given to you In terms of capital T, you need to find out these three things Equation of shm, the time taken to go to the extreme The extreme where it was trying to go And time taken to come to the mean position Time taken to come to the mean position Alright, apart from T you have capital A that is also given to you And anything else you require? No, that's it Find out Can you tell me what is the value of 5 in the equation of shm? Have you calibrated that as at least? What is 5? Sine of 5 is x by A In this case, it is minus of root 3 by 2 Now, without even solving the question Can you tell me anything about the 5? Between which two values the 5 should be? Can you tell me? Can you guess between what all the value of 5 should be? It is 0 here, pi by 2 here, pi here, 3 pi by 2 there And then 2 pi here Between pi and 2 pi Pi and 2 pi is a big range A smaller range will be Between pi and 3 pi by 2 Or between 3 pi by 2 and 2 pi Which one? It is here and is going this way Between pi and 3 pi by 2 Once you know that, it becomes so easy to find out the angle of 5 Okay So now can you tell me what is the value of 5? It will be pi plus pi by 3 That is 4 pi by 3 You can check that Sine of 4 pi by 3 is minus root 3 by 2 Okay So omega, how will you write in terms of capital T? What is the value of omega? 2 pi by omega is a time period So 2 pi by time period is omega Okay So you can substitute omega and 5 You get equation of the SHM This is part A Part B, how will you do? Just tell me how will you do that? Take into go to extreme Any idea, how will you proceed? Equation is in front of you All you have to do, put the value of X to be How much? Value of X to be You have to substitute the value Put X equal to minus A You are going this way This is minus A, X equal to minus A Just put it over here and get the value of T Okay, it's direct It's from your textbook, salt exercise This Alright Now If sine of 2 pi by capital T T This is equal to minus 1 Sine inverse minus 1 is 3 pi by 2 Okay So 2 pi by T Into small t plus 4 pi by 3 Is equal to 3 pi by 2 Okay So from here, when you simplify this You will get T T is equal to Small t is equal to capital T by 12 This is how you solve it Fine So part C is the homework So if you are really interested You can do it during the break time also As in now After the break, we will continue with a new topic Fine So let's come back after the break now We will meet at 11.27 am Come back in time