 So, they probably have just started the SHM chapter probably. You are in damping, so why is someone is asking what is SHM? Anyways, so let us study this chapter, oscillations, SHM is simple harmonic motion, okay, fine. So this chapter, which is going on in your school, we are going to start this chapter only, you may not be aware, but in other batches I have started gravitation, the gravitation and some other batch I have started the kinetic theory of gas, so all three batches now have different different chapters going on because your schools are teaching different topics. Okay, so oscillation, now oscillation is a topic which deals with, write down, it deals with a special kind of, special kind of motion, okay. So, you know, there can be many different kinds of movement a particle can have, but in this movement, this kind of movement is observed many places, many a time that is where we are discussing this kind of movement exclusively in this chapter, okay. Now, if we just look at this chapter at a higher level, you will see this chapter has two parts, part one of the chapter deals with the kinematics, kinematics of oscillation, as in we will discuss the movement only without getting into what is creating the oscillation, how oscillation is getting created, we don't care in this part, in this part we are just analyzing that okay, fine, if this is a displacement, what is the velocity, if this is the velocity, how to find the acceleration without getting into how it is getting created, fine. Just like your first two chapters of the class 11th, then the next part of the chapter, part two deals with the dynamics of oscillation, which is similar to the Newton laws of motion chapter and work by energy chapter, here we will discuss about what kind of force is required to create oscillation, right. So, we will discuss about the cause of the oscillation and we'll also discuss how much is the energy involved in the oscillation as such, okay. So, part two has these two things and part two is a very big thing because here you'll see many different kinds of numericals, when we discuss part two laws of motion, work by energy, rigid body motion, everything will come crashing down and you'll see, you know, humongous varieties of numerical, only when you expose yourself to the concrete level exam preparation, but if you just focus on the school level, then yes, part one and part two, they are similar in nature for school preparation. Then you have a small part three of the chapter that deals with non-ideal oscillations or you can say realistic cases. In part two, when we'll discuss about the cause and the energy, we never assume that some damping force is there because of which the oscillation will die down slowly. We'll assume that it'll keep on happening forever, okay. In part three, we will discuss some of the scenarios in which it's a real scenario, okay, wherein damping is present. Sometimes you have oscillation that is forced, okay. You're forcing something to oscillate. For example, you might have sat on a swing, okay, when you sit on a swing, you yourself oscillate and because of your oscillation, the swing also starts swinging, okay. First you, you know, move your upper body like that in a particular fashion, then you, when you first try to adjust your frequency of your own movement, when you sit on the swing, once you get the rhythm, then you'll see that swing is responding to your movement. Have you ever observed that when you sit on a swing, you are moving your body so that the swing also start oscillating and after some time, what'll happen, you'll get the rhythm, you will oscillate yourself with the natural frequency of the swing and that's where the amplitude of the swing increases more and more. Have you observed this? Have you ever been on the swing or mobile phone only? Same style dude, okay, fine, some of you did experience it. All right, so this is the overall, you can say, structure of the chapter and, you know, in our textbook, again, I'm highlighting this fact that first four or five chapters are the basis of the entire thing to come, not only just in 11th, but in the 12th also, until work by energy, you have to be thorough. Then same laws of motion, work by energy, kinematics, these three things will come in every chapter. Oscillation, you can see kinematics, laws of motion, work by energy is there inside the kinematics, inside the oscillation. Then same three things are there inside rigid body motion. Same three things will be there inside the waves chapter also, okay. Then you'll go to class 12th, same three things, kinematics, work by energy, laws of motion will be there in the electrostatics, will be there in magnetism, okay, will be there in nuclear physics. So these chapters won't leave you, better you master them, okay. So today my plan is to discuss more and more about the first part of the chapter, which is kinematics of the oscillation, okay. Now, before I even start this chapter, we need to be very clear with what is periodic motion, what is oscillation and what is simple harmonic oscillation, okay. So can you tell me what is periodic motion? What is a periodic motion? Only Auro knows it and Anusha knows it and Pradiv and Aditi and Siddhach, okay, that's it. So periodic motion is a kind of motion in which the motion repeats itself after every intervals of time. If there is a movement that repeats after every some interval of time, every fixed interval of time, then you'll say it is a periodic motion. Now repeats means what? What is repeating? Motion repeating means what? Motion is repeating means what? What thing should get repeated? Displacement should get repeated, velocity should get repeated, acceleration, what should get repeated? What should get repeated? Everyone, repeat of all motion variables. And motion variables will be like displacement, position, velocity, acceleration, rate of change of acceleration also you can say and rate of change of that also you can say. You can keep on listing all the variables. Everything should get repeated, everything, then only it will be a periodic motion. So can you guess a periodic motion? Some periodic motion. I coming here and teaching you every week, is that a periodic motion? Is that a periodic thing, right? It's a periodic thing every week. How it is not a periodic thing? Yeah, in other institutes what happens is classes doesn't happen in time, even teachers get changed after every few weeks, but in center there'll be a periodic motion. So same teacher will be there every week. No, I teaching different things will be something else, but I come now, my coming here is a periodic thing. All right, now this is we can say just a explanation of what is periodic. Let's talk about movement. Can you guess, can you give me an example of some periodic movement which you observe? You observe heartbeat, footage, heels, comet, something which you observe pendulum, pendulum is a periodic motion, fan rotating is a periodic motion, very good, very good. What else? Revolution of earth on its own axis and around the sun both are periodic motions, right? So there are many examples of periodic motion. In fact, to tell you frankly, every machine, every machine, you can just think of any machine, every machine runs on periodic motion, every machine, okay? So that is a reason why the periodic motions are very, very important thing to study. If you do not understand periodic motion properly, you'll not understand how machine works, okay? This is the periodic motion. Now, what is oscillation? Oscillation or harmonic motion, what it is? Hmm, everyone, it is basically to and throw motion about a fixed point. Understood? This is the, this is the harmonic motion or oscillation. So is earth revolving around the sun? Is it a oscillation? Is it oscillation, everyone? It's not oscillation, right? Because it is not to and fro. What is to and fro? To and fro means, suppose this is a point, you go away from the point, then you come close the point, you go away from the point, and then you come close the point. This is what the to and fro motion means, okay? So can you give examples of the oscillations now? Pendulum is one. Pendulum, I am writing. What else? Molecule vibrating, okay? No, no. I mean it is, but then give me a better example, okay? Swing is a pendulum only, don't you think so? Same thing you're telling. Okay, suppose if you have a bowl like that and you put a mass over here, this mass slides down, moves up over here. So like that, is it a to and fro motion or not? You keep on doing like this. Is it to and fro? Is this oscillation or not? This is oscillation and is this an oscillation wherein you have a mass, you have a spring like this, you push this mass against the wall and this mass starts doing like that. This is also to and fro, okay? Then there are other examples. Suppose there is a beaker of water on which you have a wooden log which is floating, floating in the water. You push the wooden log, slide it down. The log will start doing up and down movement like that. This is also oscillation, right? So there are many such examples in which to and fro movement happens, okay? So this is the oscillation or this is the harmonic motion, all right? Now the question is, what is simple harmonic motion? Any guesses? What is simple harmonic motion? Simple harmonic motion is simplest of all harmonic motion, okay? Of course, this is a vague definition, but then let us see what is the proper definition of it by using some analysis, okay? We will analyze what is harmonic motion first and we will simplify whatever comes out and tell there that listen, when you simplify whatever comes out, it will become simple harmonic motion. Let us see, okay? So we are not talking about the all periodic motion. By the way, are oscillations periodic motion? Are oscillations periodic motion? Oscillations are periodic in nature. They happen again. It repeats after every time, after sometime, but all periodic motions are not oscillations, okay? So oscillations are subset of periodic motion, fine? Now since we are talking about oscillation, which is to and fro, let us take an example of to and fro motion on a straight line, okay? Just to keep our life simpler, we will assume like this. So it is to and fro. So there has to be a fixed point. Let us call that fixed point to be O because oscillation should happen about that fixed point. So that fixed point is O, okay? And then it is to and fro. So there will be a limit up to which object will go because it has to come back, right? It cannot just keep on going, going, going. So there will be some point, let us say, this point is M. M is the mass, okay? P. This point is P. And similarly, same thing will happen on the other end. This point is Q, okay? So their entire oscillation, entire oscillation happens in this straight line about that fixed point O. The object, you can say, starts from anywhere actually. Right now I am assuming that object starts from O. So let us name these points first, then we will proceed. O is called mean position. For all oscillations, this is called mean position, whether it is simple or not. P and Q. Do you know what is the name of P and Q? What is the name of P? What is the name of Q? Extreme, correct, extreme positions. These are the extreme positions. Now the maximum distance, the maximum distance from mean position an object can go up to is OP. What is OP called? OP is what? Distance OP. O to P distance. Do you all know it is called amplitude? It's amplitude. Clear to everyone, okay? Now let us talk about the entire one cycle of the movement. So first the object will go from O to P. This is the first movement, O to P. Then it will come back to O. Is that one complete cycle? Is that one complete cycle? Why it is not? Which motion variable is not getting repeated? No. Which motion variable is not getting repeated when it comes to O? The velocity is not repeated, right? See acceleration and displacement, both are repeating but velocity is not repeating. So you can't say this is one cycle, okay? So it has to go further like that, like this and then come back to here. So total, how much distance it has travelled? If amplitude is A, total distance travelled in one cycle is how much? In one cycle, the distance travelled, a distance covered is 4A. Is it true if it starts from P also? Suppose the entire SHM rather than starting from O, it starts from P. Then also it is true. This is OQP. Suppose it starts from P. Then in one cycle, what is the distance travelled? It goes like this, goes back like that. The distance is 2A and then comes back like that. The distance is 2A. So total distance is 4A again. So it does not matter from where you start, the distance travelled in one cycle will always be 4A. Clear? Okay? Now we need to give some mathematical form to the movement of the particle, otherwise we will not be able to analyze it. Okay? So let us see what is happening. What are the kind of observation? Then based on the observation, we can see mathematically how can we write the equations. So let us analyze the movement. O to P. Let us say O is origin. Okay? O is origin and I am tracking these variables. I am tracking the x coordinate, the velocity and acceleration. You just have to tell me whether something is less than 0, greater than 0 or equal to 0. Okay? From O to P, throughout the movement, the x coordinate, greater than 0, less than 0, what it is? Throughout O to P movement, x is greater than 0. All of you agree? x is greater than 0. Everyone? Now O to P velocity greater than 0, less than 0. Everyone? Velocity is in this direction and that direction is positive direction. Is it greater or less than 0? You need to better answer whatever I am asking. Greater than 0, right? Clearly. Now tell me the acceleration. When you go from O to P, the particle should accelerate or decelerate. When you go from O to P, it has to slow down or it has to increase its speed. What do you think? Decelerate? So acceleration is in which direction? Right hand side, left hand side. Left hand side. So, the acceleration is less than 0, greater than 0, less than 0. Okay? So, this is O to P movement. Now, let us see P to O. P to O, tell me x, V and A. x is greater or less than 0. P to O, when it comes back, x is greater than 0, right? Velocity? Velocity is less than 0, left hand side. What about acceleration? Acceleration, greater than 0? Everything is greater than 0. When it is coming from P to O, acceleration is on the left hand side or right hand side? Which direction the acceleration is? Left hand side is positive or negative. You cannot change the sign convention in middle. Okay? So, the acceleration is also less than 0 from P to O. Okay? Now, let us look at the other two parts. O to Q, O to Q, x, V and A. x is what? O to Q. All of you, less than 0. Velocity is, what about velocity? Less than 0. Acceleration, greater than 0, right? It has to be on the right hand side. It is decelerating. So, the acceleration is on the right hand side. Okay? This is O to Q. Now, talk about the last thing, Q to O. Q to O, x, V and A. Tell me, x is less than 0. Okay? See, you need to participate in these things. Otherwise, when things will become tricky, you will be like, oh, I did not pay attention at this start. So, the excuses are ready with you. Right? Better you pay attention and participate. Otherwise, things will become tricky very soon. Then you will not get that easily. So, x is less than 0. What about velocity? Greater than 0. Acceleration is also greater than 0. Greater than 0. Okay? Now, every oscillation that happens, should follow this or not? Every oscillation in the under the sun should follow this, right? So, if every oscillation follow this, then can you see any pattern here? Can you tell a pattern? Based on pattern will create a mathematical formula. Are you able to see any pattern? In entire motion, there is a pattern which you need to identify by using these things. What is that? Yeah, some of you already know the answer because it is going on in your school. Okay? Others, what pattern do you observe here? Can you see that all after all 4th 4 path all over it repeats? Yeah, that is that is not the pattern. I am asking what is the pattern among the motion variables x, v and a are they related in some form? As in, is there a pattern among them that when one changes, the other also changes or this is the, you know, some sort of nobody else, you're not able to identify the pattern. Now, tell me in all 4 paths which you have mentioned, for a mean, extreme, v and a are the same sign. Yes, Aditi, that is not the pattern actually. Okay? Mean to extreme, v and a mean to extreme. No, it is o to p, you can see v and a, they are not of same sign. You can see v is greater than 0 and a is less than 0. x and v, same sign. p to o, they are not of same sign. They are not of same sign. No, you can't be specific about it. You, no, no, no, that is not the pattern. Pattern should be valid for all 4 paths. Okay? All 4 paths. The pattern is, can you see that the x and a of opposite signs always? Can you see this? All of you agree? Okay? So this is the pattern that x and a, if they are of opposite signs, then oscillation will happen. Period. Okay? Clear? Now, looking at the oscillation, can you tell me what is the, what is the velocity at point p? What it is? Velocity at point p is how much? Does the object need to stop at point p or not? Point p, object need to stop or not? Simple question. Why object has to start? Why object have to stop at point p? Because it is changing its direction. So it can't change direction without going to 0. All right? So velocity at p and velocity at q is 0 for all oscillations. Doesn't matter simple or not. Extreme position, velocity should be 0. Okay? Can you tell me based on these, whatever you have written here, don't try to recollect from your memory final answer and least interested. I am interested in have you, do you have capability to think from whatever is written and then answer? Okay? So can you analyze this and tell me where the velocity will be maximum and tell the reason for it? What is the reason? Again, Priyam, you are not recollecting from whatever we had done. You are using something which you already know. Did I teach kind energy, potential energy? Okay? So from p, the object is addressed at p, it starts accelerating till 0.0. When you go a little bit this side, is it accelerating or decelerating? From o to q, it is decelerating or accelerating? Everyone, you can see velocity and acceleration are of opposite signs. So decelerates. So till o, it's accelerate. So velocity at o is maximum. So at mean position, velocity is maximum. This is very clear for every oscillation. Doesn't matter simple harmonic or not. Okay? Now can you tell me where the acceleration is 0? Acceleration is 0 where? Acceleration at o is 0? At o is 0? Why? Why it is 0? Because it is changing its sign. Okay? Because it is changing its sign. So acceleration at o is 0. Okay? So without even knowing any formula, any equation, just by observing the path, you can tell all of this. You don't need to know anything else. Getting it? Now, I was thinking about something, forgot. Okay, I'll come back if I remember. Anyways, so the v0 is maximum at o because from point p to o, it is accelerating. Acceleration will increase its velocity and immediately after it goes from o the other side, it starts decelerating. So that is why at o, the velocity is maximum. So if acceleration and x have to be in opposite direction all the time, then can I say that acceleration is proportional to? Acceleration is proportional to negative times x to the power n. Can I say that? Everyone? I have to only make sure that a and x, they are of opposite signs. I can say this, n is some number, something. 1, 2, 3, 3.5, 4.5 can be anything. Do you all understand this? So if acceleration is equal to minus some positive constant x to the power n, this satisfies the criteria that a and x, they are of opposite sign. Do you all agree? Type in quickly. This is not correct. There is a small mistake. Can you tell me what is that? Think about it. If you are able to tell that, it will be good. What is the mistake there? How can you correct this formula? C is positive. Yes, C is positive. I am saying C is positive. It's not negative. Don't worry about that. Be positive. I am saying C is positive. Then also there is a mistake in this formula. None of you are able to? No, x can be negative. Why not x cannot be negative? What are you telling? See, when you have analyzed the path, what do you mean? Correct. So if n belongs to even number, even number, then if x is negative, then a and x, both will come out to be negative or not if n is even. By x is negative, so n, x to the power n will become a positive number. Do you all understand that? It will become positive if n is even. So if I say a is equal to minus c, x to the power 2. Now, if x is equal to minus 2, if I write x equal to minus 2, I will get a is equal to minus of 4 times c. So you can see even a is negative. x is also negative, a is also negative. But that is not how SHM equation, that is not how oscillation equation should be. Do you all understand this? Everyone understands that? Whatever I just did. So hence, the correction is a is equal to minus c to the power x, c, x to the power n, where n belongs to odd integers, odd integers. Getting it? Okay. And what? c is greater than 0. So I will be extra careful with c. I can never take c to be negative, otherwise SHM won't happen. So just to be extra cautious, you know, how will the right constant? I'll write constant as omega square that way. And then I will be always be conscious that this constant, if I write it as square of something, it can never be a negative constant. Okay. So there comes this equation, which is your oscillation equation. This is equation for harmonics. All right. So simple harmonic motion is simplest of all motions. n belongs to odd integer. n can be 1, 3, 5, 7, and like that. So when n is equal to 1, the equation becomes this omega square x. And this is the simple harmonic motion SHM. The simplest of the harmonic is simple harmonic motion, which is when n equal to 1. Clear to everyone? So this is the start of the chapter. This is where the chapter starts the simple harmonic motion. We are going to take this equation and we are going to analyze it in the entire chapter. Okay. So a is equal to minus omega square x. So from here, you can see that acceleration is such that it always, always tries to push the object towards mean position. You can see when x is greater than 0, it tries to decrease x. Acceleration is opposite of it. When x is less than 0, it tries to increase x by bringing it closer to 0. So every time it tries to bring near the mean position, this is what the SHM acceleration is. And the, of course, this thing we will study, force will study later on, but since we have got the formula for the acceleration, the force is mass same acceleration for a mass. So if there is a mass which is executing SHM and omega square x, this force, what it tries to do? Tries to restore the position of object on the mean position. So to perform SHM, to perform SHM, we need restoring force, which is like this. Okay. Very clear. So we are going to discuss about the force, energy and all that later on. Right now, I'll again skip this concept of force and come back on the acceleration. a is equal to minus omega square x. Now tell me, can I use equation of motion over here? v is equal to u plus a t, s is equal to u t plus half a t square. Can I use that over here? Everyone, a really good poll. Tell me, can I use equation of SHM, sorry, equation of motion in the case of SHM, like v is equal to u plus a t, s equal to u t plus half a t square. Can I use it over here? We have spent a lot of time in kinematics. Done. Only one person hasn't taken the poll. What it is? Should I find out? Still. Okay. So this is the poll results. There are two people, two students who are saying yes. The answer is no. We cannot. Okay. Why? Because the expression is not constant. These equation of motion were derived. We have gone through all the derivations, right? s equal to u t plus half a t square v is equal to u plus a t. All that is derived by assuming acceleration to be constant. So you cannot use it over here. The reason is, acceleration is not constant. It changes with x. Okay. It is a not constant acceleration. Because of that, we can't use equation of motion. All right. So if we can't use equation of motion, how to analyze this? How to analyze this movement? SHM movement, how will you analyze? Can we use differential form? We can use differential form. Of course, that you can use anywhere. Differential form. But it makes sense. It makes sense to derive equation of motion using calculus for SHM as it is a common type of motion. See, even if acceleration is constant, then also you can use calculus to analyze everything. Why you have s equal to u t plus half a t square v is equal to u plus a t. Why you had equation of motion? Because when acceleration is constant, that is a very common scenario. That is why you are like, okay, fine. Let us derive equation which I can use without bothering myself with calculus. Similarly, this SHM movement is also very common. So that is the reason why I do not want to bother myself with the calculus every time. So I am going to use calculus just like I have used calculus to derive constant acceleration motion equations. Similarly, I will use calculus to derive equation of motion for the SHM kind of movement. So let's try to do that. Do not use s equal to u t plus half a t square v is equal to u plus a t and v square equal to u square plus 2 as for SHM ever. Remember that we are going to derive equation of motion for SHM. So acceleration is equal to minus of omega square x. How can I solve this difference? By the way, this itself is first equation of motion. Don't you think so? a and x, this is a relation between a and x. So isn't like an equation of motion? Isn't like an equation of motion? All of you, right? This is our first equation of motion. First equation of motion is this only. What is the second equation of motion? How can, I mean, this is a relation between a and x. Can I get a relation between v and x? Can I do that? How to convert this relation between v and x? How to do that? So get the equation, all of you. What do you write a as? What do you write a as? dv by dt. If you write a as dv by dt, you have three variables. You can't integrate that. Preum understood that. See, you brought in t over here. Better is, good. Aura got it. Oshe got it. Praktu got it. Preum got it. Good. You guys have become intelligent. So a is equal to v dv by dx. This you have to use. This is equal to minus omega square x. Okay. Now integrate and get the answer, all of you. When you integrate what you put limit as at x equal to a, what is the velocity? 0. At x equal to a, velocity is 0. And at x equal to x, let's say velocity is v. So can you find the velocity now, everyone? Anyone got it? Just you type in that you have got it. You don't need to type the answer. So how many of you got v square is equal to minus of omega square x square minus a square. How many of you got this? Divide by 2 will come v square by 2 x square by 2 and 2 will get over. So from here, v is equal to omega root over a square minus x square. This is the second equation of motion. First one is a relation between a and x. Second one is between v and x. These equation of motions are better than the constant acceleration equation of motion. Because in constant equation acceleration, constant acceleration equation of motion, you do not have velocity and x separate. Always a is there in the expression. In all three equation of motion, a is there, acceleration. Here a and x, one equation, v and x, another equation. Now time factor we have to bring in t. So can I get x in terms of t? Can I get x in terms of t? How should I proceed? How should I proceed? Write v as dx by dt. dx by dt is omega times root over a square minus x square. Then what you do? You integrate this. You guys have done all this derivation already. This is equal to omega times dt. Now you integrate this. At t equal to 0, where the object is, what should I write here? x equal to what? At t equal to 0. Why? Why t equal to 0, x equal to 0? SHM can start from anywhere? Do you all understand? SHM can start from anywhere. Let's say x1 or let's say x0. Why? SHM need not start only from the mean position. Come here. That's one. SHM can start from here also. It need not start from the mean. At t equal to 0, what is t equal to 0? t equal to 0 is the time when I have started counting the time. I can switch on my stopwatch at any moment. So at t equal to 0, x equal to x0 and t equal to t, x is x. Now do anyone know what is the formula for integral of this? Do anyone know? If you don't know, it's okay. It's nothing. It's just a formula. When I tell you now, you know. If you know it, that's good. If you don't know it, that is also good. Nobody will become intelligent by just knowing some formula. So I'll tell you now the formula is sine inverse. Have you seen inverse? Sine inverse? Have you seen ever? Okay, good. So at least sine inverse you have seen. This x0 to x, this is equal to omega into t. So when I put the limit, it will be sine inverse x by a minus sine inverse x0 by a, this is equal to omega t. Okay. Till now, all of you clear? Everyone? Type in, is it clear till now? So I can say that sine inverse x0 by a is some angle phi, some angle phi. Sine inverse is an angle only. Let's say x sine inverse x0 by a, a is some angle phi. Then I can write down sine inverse x by a is equal to omega t plus phi. Okay, or x by a is equal to sine of omega t plus phi or x equal to a sine omega t plus phi. This is our third equation of motion. Relation between x and time, x and t. This is a relation. Right? Now tell me what is the value of phi when x0 is a, what it is? Sine inverse x0 by a is equal to, when x0 is a, it will be sine inverse 1, sine inverse 1 is pi by 2. Okay. And when it is minus a, then what it is? Sine inverse minus a by a, sine inverse minus 1 is 3 pi by 2. All of you? All of you getting this? Okay. Now listen to me carefully. Here, according to this, the entire SHM can be seen as if fluctuation of angle from 0 to pi by 2, then it reaches pi here. Then this is 3 pi by 2 and then comes back to 2 pi. If you look at phi, sine inverse x0 by a, that is how it is, okay? From 0, it grows to pi by 2. Suppose it starts from here, then phi is pi by 2. It starts from here, but instead of going in positive x-axis, if it goes in the negative x-axis, then phi is not 0. Phi is pi, okay? It starts from here, phi is 3 pi by 2. And 0 and 2 pi both are same only because 2 pi is a cycle of sine. All of you understand, right? So this is the equation of SHM x versus t. Now here, now again, you know, because these are important things, a is equal to minus omega square x. We have v is equal to omega root over a square minus x square. These two equations, it does not matter, doesn't matter t equal to 0 where the particle is. Time is not in the equation itself, all right? But when it comes to x versus t, x equal to a sine omega t plus phi, you need to find out phi, okay? Where in phi is sine inverse initial position of x divided by a. And you know, when it is at the mean, when it starts from the mean position and goes in the opposite direction, then sine inverse 0 have to take pi, not 0. But anyways, we'll see in the numericals. Here, it matters what is phi when you find out x, where x is at a particular time should depend upon where from where it has started, right? It might have started already from the mean position. So x is already a when t is equal to 0. So you can't say x equal to a sine omega t always, right? So can you tell me velocity as a function of time, everyone? How to get velocity as a function of time? You can differentiate x dx by dt. This is equal to what? A omega cos of omega t plus phi. You could have directly written the value of x over here. I mean, you simplify, it will come out to be this only. All right? Now, can you get x version as a function of time? A is equal to what? You can differentiate v. You can either do that or you can substitute x over here. x is equal to a sine omega t plus phi and substitute there. But you can differentiate also. So this will be equal to minus of a omega square sine of omega t plus phi. So you can see that a into sine of omega t plus phi is x. This is minus omega square x. So you can play around. I mean, they are connected in a way. So how many equations of motions you have? One, two, three, four, five. Why you need concentration three equation of motion? We don't need them. All right? So make sure you are using these equations of motion when you are analyzing the SHM. Okay? Clear to all of you. We have, I think, couple of more minutes. What we can do is, let's do one thing, a small question. You have to plot acceleration versus x plot x on the x-axis. And second plot is velocity versus x. First do acceleration versus x. All of you. Did you plot the acceleration versus x? Anyone? This is x. This is a. a is equal to minus omega square x. How will you plot, everyone? What you get the answer as? Is this the answer? All of you? Is this the answer? All of you getting typing? Yes. So now quick, we have to end the class quickly. No, this is not correct. What is wrong here? Something is wrong. No, Anusha, not that. You can check after the class or ask doubts after the class also. Right now focus here. Something is wrong in this graph. What is wrong? What is wrong? You've heard about physics. The moment I asked about the graph, your focus was mathematics. Can x be more than a? Can x be more than a in SHM? So you don't get line. You get line segment. Okay? This is what you'll get from here till here. That's it. This is a graph between a and x. All of you understand this? x can't exceed a. So this is x equal to a and this is x equal to minus a. There is no point beyond that. Now can you plot v versus x? Everyone do this. This is the last thing for today. Velocity versus x. Is it magnitude of these velocity versus x? Oh, sorry, not magnitude. You take with sign. Sign is included. This is x. This is velocity plot the graph. All of you do you understand? There is no point on the graph beyond x equal to plus a and x equal to minus a. You can't exceed. You can't go in this side or that side. Do you all understand that? So your plot should be between these two lines. All of you type in. Do you understand that fact? Plot should be between these two lines, right? And at x equal to a, the velocity should be 0. x equal to minus a also. Velocity should be 0. So two points are here only. At x equal to a, velocity is 0. x equal to minus a, velocity is 0. These are the two points on the graph. Can you find out some one more point on the graph? These are two points that will be there. One more point on the graph. Can you find out v is equal to omega times root over a square minus x square. So at x equal to 0, velocity is omega a. Can it be minus omega a also? Can it be minus omega a also? It can go positive side as well as negative side, right? So at x equal to 0, it can go there as well as there, right? When it go to a right-hand side plus omega a, when it starts going left-hand side, minus omega a. So it can have both at x equal to 0. So this is minus omega a and this is plus omega a. So you have four points on the graph, right? Now tell me one thing. You have an SHM going on. You take any x over here. At this x, according to this formula, the minority of velocity will be this only. But at that x, will you have two velocities, one forward, one backward, one while going this side, one while coming back, will you have two velocities or not? Everyone, right? So one will be positive, other will be negative. So as you go towards the mean position, magnitude of velocity will decrease. So you will now get a clear picture that the plot will be like this. My drawing is not as good. It will get something like this. For every x, you have two velocities. One is positive, other is negative. Same magnitude for every x. So you have two velocities, this and that, this and that. When x equal to a or minus a, you will have zero velocities. So just so that you know, this is ellipse. You will get an ellipse when you plot v versus x. All right? Fine. So this is just an introduction of the beginning of the chapter. Next class, we are going to talk about some questions on these equations that we have derived today. And then we are going to start the, start what? Start the dynamics of the SHM and talk about what kind of force create SHM and how to find out the time period of the SHM. That is the most important thing about SHM, time period of oscillation. If you know the time period, you can tell many things. All right? Fine. And just one final thing, one final thing I should have pointed out earlier. Now you understand why looking at this equation, this one, now you understand why we write omega square over here, that omega sits here. Okay? It nicely, that omega square becomes the sign of omega t. Fine. So that's it from my side. We will meet next week with simple hammering motion chapter. Bye for now.