 Now, we are talking about the causes of the SHM, what causes an SHM? What kind of force creates the SHM? All that discussion we will do here. Now, let us first analyze the oscillation, what happens exactly? You have this point O, you can see that acceleration is always towards a mean position. When you go from O to P, acceleration is this way, P to O also, acceleration this way and over here, accelerations are this way. So every time, acceleration should be towards the mean position, right down, acceleration must be towards the mean position. Now, if acceleration is towards the mean position, the force is also towards the mean position because force is equal to mass times acceleration, mass is just a constant. So we need, we need a force that pushes the mass towards the mean position always. So when you go on the right hand side of the position, the force pushes towards O as in left hand side. When you go left hand side away from the mean position, the force pushes you on the right hand side towards the mean position. So every time, the force is trying to bring the mass at the mean position. So the force is, force is trying to restore the position of mass at O, at the mean position. Do you know what it is doing? So this force is a special kind of force which does like this. The name of this force is called restoring force. I hope this is clear to everyone, all of you clear. Now how much is the force at the mean position? At mean position, the value of force is, how much? It is zero because acceleration is zero, so force is also zero. So that is another way to define the mean position. Mean position is an equilibrium point and to tell you in a very analytical way, SHM is nothing but a study of stable equilibrium. It was in stable equilibrium at O, it displaced it slightly, it comes back. So the thing is that even though there is a force that tries to restore it on the mean position, it never stops at mean position. Why? Due to inertia, due to inertia, even though force is equal to zero at O, it keeps moving. So what will happen? The force is trying to push it towards the mean position and as soon as force thinks that the object has reached the mean position, force stops that, yeah, I have done my job. But then it has mass, it has velocity, so it keeps moving away. Then force will be like, oh, it is going away from the mean position now. So force will now start acting in opposite direction towards the mean position. Now as soon as the object reaches mean position, force will withdraw itself, force will become zero. But then again, due to the momentum it has, it moves forward. So it keeps on doing like this and that's how the SHM gets created and this is the cause of the SHM. Okay? Anyone has any doubt? No doubts? Okay. So we need a restoring kind of force and acceleration is omega square x minus of omega square x. So force is equal to minus of m omega square x. So we need a force, we need a force that gets developed due to the displacement away from mean position. When the object is at mean position, all the forces were adding up to zero. But as soon as you displace it a little bit away from the mean position, a force which looks like this should get developed. What is this? Looks like means the force should be proportional to x, okay? Force should be proportional to x and force should be in opposite direction of x. That is why there is a minus sign. Okay? Suppose you displace the object this way, the force should be in opposite direction of that displacement, fine? So the simplest possible scenario for the SHM is the spring block system, all right? So we will talk about the simplest scenario is spring block system that to horizontal. So we will first analyze this and then I will tell you the steps which you should use to solve all the questions related to it. So suppose this is a scenario, all of you draw this with the block spring constant k, this is mass m, all right? This is smooth. So how will you identify the mean position in this system? Everyone, where is a mean position right now? Mean position net force should be zero. So how much is the extension the spring at the main position? Everyone, when the spring extension is zero, okay? That is a mean position, net force is zero. So mean position is right now itself. Now what I will do is that I will displace the block by a distance of x and then release it. So what will happen to the block? Where it will go? It will go back towards the mean position like this. So can you tell me what is the amplitude of oscillation? It develops an oscillation, is the amplitude of oscillation x only? It is x because at extreme position velocity should be zero and it starts going towards a mean position. It never crosses this point, x becomes your amplitude itself by initial displacement, okay? Now suppose you displace by x, the force that get developed is how much? How much force get developed in the horizontal direction because of displacement? Spring force, if a spring get extended by x, you do not know. How much is a spring force? Everyone, Harsha, what is a spring force? Everyone, Aryanjha, what is a spring force? Harshani, Pranavaditya, Harshani, Pranavaditya and what is that name I have taken? Aryanjha, answer. So the force is Kx, okay? Kx, x is this way but force is in opposite direction. If you write in a vector form, it will be minus sign over here. If this is a force, acceleration is minus K by M times x. Now if you compare it with equation of SHM, you will automatically get omega to be under root of K by M. And if this is omega, time period is 2 pi under root of M by K. So we have got time period of oscillation. I hope this is clear to everyone. Anyone has any doubt? Type in quick, else I move forward. Now we are going to discuss something very important because I am going to talk about the steps of solving problems, okay? Write down steps. Don't skip these steps for at least 40 to 50 questions. You will never learn to skip a step by skipping it. You will only learn to skip the steps in the exam if you are following all the steps while practicing, okay? Steps of problem solving. As in what kind of problems I am talking about? Problems in which you are asked to find out the time period of oscillation, the frequency of oscillation for a system and all those things, fine? They are very common. Most of the questions, 80% of questions fall under this category on this chapter. Write down. And foremost thing is identify the object that is performing SHM. You should know which object you are talking about. I mean it will not be as simple as spring block system all the time, okay? It can be different types also. Then is identify the path of SHM, whether it is doing horizontal SHM, vertical SHM, inclined plane SHM, you need to identify that. Identify the path of simple harmonic motion, okay? Then what should I identify everyone? What is the next thing to identify for an SHM? Identify the mean position, correct? Identify the mean position. How you identify the mean position? At the mean position, net force is equal to zero along the SHM path. Along the path of SHM, you need to find out at what location net force is zero. It may sound very simple and straightforward, but sometimes it is not. For example, if I tell you that I am constantly applying a force on this block F, F is a constant force, let's say. Then at mean position, spring is not in its natural length. At mean position, net force should be zero. So F is equal to KX0, you have to say that at a mean position, spring is already extended, net force should be equal to zero. This force plus a spring force should be zero, okay? But since we have taken a simpler system, I didn't include that constant force. But we will see that in some of the examples. So you identify the mean position and then what you do is that displace, displace the object by a small distance X away from mean position, okay? Along the path of SHM, your X is amplitude automatically. As soon as you release it, it will run towards the mean position. Fifth, you need to then identify net unbalanced force. See, you don't need to mug up all of these, wake up in the morning and start mugging up step number one is this, two is this, three is this, no, no, no. Even if you ask me to write it again, I may write it in a different manner. All you have to do is that align your thought process like this. When you solve, let's say, five or 10 questions, initially, look at these steps one by one, follow it. And then next 30, 40 questions, you'll automatically follow these steps. And after that, you don't need to remember it. Automatically, you will be doing it yourself. Identify the fifth step is identifying net unbalanced force due to the displacement. This is very important. Okay. What happens is that before you displace it, net force is zero. Even though there are forces, let's say external forces are there, they were all adding up to zero. Okay. So once you displace it, you need to find out due to the displacement, how much force is coming up. The extra force that comes because of a displacement is the net force. Because earlier anyway, it was zero. Okay. It was like earlier you had five, five minus five was the net force. Due to the displacement, some extra force is coming up. So your net force is this extra force only. Earlier anyways, net force was zero. So don't start calculating all the forces again one by one. You just need to focus on how much extra force is coming because of the displacement only. So we'll take examples. We'll discuss what does it mean actually. Then if it is an SHM, net force must be towards the main position. Okay. Main position and proportional to the displacement X. Fine. This is a must. If you displace it slightly and the object keeps on moving in the same direction once you release it, it doesn't come back. Then it is not an SHM. Okay. Next is compare acceleration with this and you will get time period as 2 pi by omega. So these are the steps which you should follow for all the questions consciously or subconsciously. Okay. I'll move forward. All of you have written it. Fine. So let's take few questions and see how we can use it. Suppose you have a spring block system. We'll be taking simple systems only today. Spring block system, but the spring is vertical now. Mass is M and spring constant is K. Okay. You need to follow all these steps and get the value of time period of the SHM. Okay. Try doing it yourself. That is the best thing that you can do. Don't wait for me to write down and you copy it. At the main position, will the spring be extended or not? This block is doing SHM. Path of SHM is vertical. These things are straightforward. We'll do SHM like this. Add the mean position. Let's say this is the mean position itself. Net force should be zero. So this is MG and this is initial extension is X naught. Let's say KX naught. So add mean position KX naught minus MG is equal to zero. So you'll get X naught. Basically, we don't need to do all of this, but I'm doing it so that you understand everything. Now, what is the next step? What should I do? What should I do next? Displace by X. So let's say I have displaced the object by X down here. So what is the force that comes in because of the displacement? Everyone, when I displace it by X, the force because of displacement is KX or KX plus X naught. Initial extension was X naught. Force because of displacement is KX or KX plus X naught. Everyone, KX naught was anywhere there. It doesn't need a displacement of X. Because of X displacement, KX is the only force. But if you're like, no, I'll find out the total net force. Then you can do that. Total net force is KX plus X naught minus MG. MG is also there. Then if you take KX naught, even MG is there. So this is equal to KX plus KX naught minus MG. KX naught minus MG was zero earlier itself. So net force is KX. So you could have written KX directly. That's what my point is. At mean position, anyways, net force is zero. So extra force that comes in because of displacement is the net force. So KX is the net force, which is equal to mass and acceleration. So in a vector form, A is equal to minus K by M times X. So it will be good to see that time period still remains the same as if it is horizontal. So it doesn't matter whether your spring is horizontal or vertical, the time period remains this only. Is it clear to everyone? Type in? All of you type in. Is it clear? So now let's do one more. In fact, the second part is all about problem solving. There is no more theory now. It's all numericals going forward. That's how class 11th is. Even the 12th is like that. So make up your mind like this. The main work is solving numerical. Main work is not knowing the theory. This is M. There is this wedge which is fixed. Spring constant is K. Find out the time period of oscillation for this. It'll be way better if you can do it yourself. Try doing it for two, three minutes. The question is find the time period of oscillation for the mass M which is placed as shown. Initial extension in the spring is how much? This is the path of oscillation. OK, all of you tried? Anyone about to get the answer? OK, I'll check it out. So there will be initial extension in the spring. But do I need to find out that? Do I really need that? We don't need that because at that location, the net force is 0. But then if you are very curious, you can spend some time on it to get it. This is Mg sin theta. And there will be a spring force backwards along the path of SHM when Mg sin theta and Kx, they balance out. This is what it is. X naught is equal to Mg sin theta by K. This is the initial extension. Now when you displace this object by distance of X further, the spring force is actually Kx plus X naught. But I don't care about the earlier spring force. I only care about the extra force because total force now will be Kx plus X naught minus Mg sin theta, which will become Kx only. One thing which is very important to understand in these questions, where I have seen many students make error, is that this X, people sometimes take it as extension in the spring. X is not extension in the spring. X is distance away from the mean position. So remember that. Now total force is Kx, which is equal to M into A. Exactly same thing comes again. K by M times X. So time period is 2 pi under root M by K. So surprisingly, the spring block system doesn't matter how it is kept. Time period will be always this. It doesn't even depend on gravity. Even if gravity is not there, you will have this. That is why people say that when you are in a space, you should not use the pendulum clocks. You should use spring block system clocks, wherein time period is measured by the oscillation of spring block. All right, so this is the spring block system. Now talking about the spring, do you remember we discussed about the combination of spring? Have you discussed about it? Do you remember it? I have discussed it. Combination of spring, series and parallel, do you remember? In work by energy chapter, yes, I have done it. So you can revisit your notes if you have made it. Recently, I found out some of the students, they were not making notes. They were just putting the headphone and sitting in front of the laptop and just like some serial is going on, they just listen it and then that's all. And then they don't even do the assignment, directly come and write the test. So if you don't follow some basics, why are you getting disappointed if you do not get marks in the exam? My simple question is that, okay? If you follow the basics, then if you're not getting exam marks, then I can discuss with you and find out what is wrong. But if you're doing your basics wrong, then I cannot help you. No one can help, okay? So you should make notes, not because it is a common practice, that is why I should do it. You should make notes because it's well known that if you write something down, you understand more about it. It means that you care about it. Your mind will care about it. If you just listen, don't bother to write anything or just start taking photographs of the screen. Ask yourself how many times you have looked back on these photographs of your screen. It's, I mean, forget about photographs of your screen. Have you looked back on your photos which you have took maybe one year before? When you go somewhere for sightseeing, you take the photograph. You're totally busy taking photographs, but do you ever watch it after one year? You forget about it, okay? So don't follow those practices which are sure shot failures, okay? There is no substitute for hardware. Don't look for shortcuts. That is like the first step towards the disaster. If you start looking for shortcuts. Anyways, coming back to combination of springs, the first combination is a parallel combination. How will you identify that it is a parallel combination? Parallel combination looks like this. So if I replace two spring with a single spring, what will be the spring constant? K equivalent will be what? K equivalent is equal to K1 plus K2, okay? And even this is the parallel combination wherein you have a block like this, K1, K2 and M. In this case also K equivalent is K1 plus K2, okay? Both are parallel combination only. B, series combination. Series combination looks like this. This is K1, K2, this is M. In this case, one by K equivalent is equal to one by K1 plus one by K2, all right? So just remember this and there is the spring law. Sometimes in questions you'll see that they cut the spring into half or they reduce the length of the spring. So spring law is that spring constant into length of the spring is fixed. If you cut the spring into half, the spring constant will double, all right? So these are the few things you should be knowing it. Otherwise you'll get stuck in the problems. At least the spring block system I'm talking about. Clear to everyone? Anyone has any doubt typing? So now we are again back to the problems. Suppose this is a situation, spring constant K, this distance is L. Now the surface is smooth. This is the wall which is elastic. Elastic collision will happen from that wall. This block is thrown forward with velocity V, okay? So can you look at it and tell me, will the motion be periodic or not? It's motion will be periodic or not? It goes, hits the spring, compresses it, comes back, goes there, hits the wall, comes here, hits it. Will it be a periodic motion or not? It'll be periodic motion. You have to find out the time period of it. Time period of the one complete periodic motion. But it is not an oscillation, it is not an SHM, okay? Because one part is spring block, other part is a linear velocity. Do you all agree with whatever I have written here? It goes A to B, compresses the spring, goes up to the extreme point C and then comes back. And when it comes back, it leaves the spring, moves forward with the same velocity, comes here, hits the wall and then reverses. So it has traveled AB distance, then B to C, then C to B, then B to A. And A to B and B to A, these two distances, constant speed is there, B to C and C to B. If I add both of them, it become part of SHM. How much part of SHM it is? One time period, one third, one and a half, one fourth, how much SHM has happened? This is what part of the SHM? B to C and C to B, they are half of the SHM. For full SHM, time period is two by under root M by K for any spring block system. So AB and BA, two L distance covered by speed V, so two L by V plus half of the SHM, this is the answer. This is the time period of the entire periodic motion. Now, our focus was mainly on spring block system right now because that is the most common. There are other situations also. Let's quickly take one of that today. We have 15 minutes, we'll take that. Without spring, can we have SHM? Yes, there are many, many situations for SHM. One of that is this. You have a bucket full of liquid. Liquid has a density of rho. You have a cube, which is floating on it. Okay, here is a cube. It doesn't look like a cube. So this is a cube of density sigma and side length. Side length is A, okay? When you displace it slightly, when you displace it slightly down and leave it, it start performing SHM, up and down. Find out the time period of the SHM, all of you. Find out time period of the oscillation. If you follow all the steps one by one, there is no reason why you will not get the answer. All right, Arjit and Gayatri got half. Oh, that was previous. Anyone close to the answer? Anyone close? Initially, it will be submerged by a distance of let's say X naught. And because it is floating, right? So sigma has to be less than rho. Anybody got X naught initial? How much was submerged? All you have to do is equate the gravity force and buoyant force. Buoyant force is rho volume is A square X naught into G. Okay, so when rho A square X naught G minus MG is what? Sigma into A cube G, sigma into A cube is mass. This is equal to zero. So X naught is sigma A by rho. But do I need to find out X naught? No, not required. Because X naught is corresponding to a point where net force is zero, okay? Now when you displace it by a distance of X, let's say displace it by a distance of X. Can you find out the net force because of the displacement? What is the net force because of the displacement? Everyone, do it yourself. Tell me, okay, Archit got something. Will there be an extra buoyant force because of the displacement? Because of displacement, there will be extra volume that goes inside the water or the liquid, which is this. How much extra volume goes down? How much extra liquid get displaced? A square into X. This is the extra volume that got displaced. So the extra buoyant force that get developed is rho A square into X into G, sorry. So the net force is this only, rho A square X into G. This is the net force in upward direction. This should be equal to mass and acceleration. Sigma into A cube is mass into acceleration. I'll take capital A for acceleration. Small a is the edge length. All of you able to follow this? Type in acceleration is equal to rho by sigma into A into X in vector form. There will be minus sign because A and X are in opposite direction. Can explain the second part. Where is the second part? This one, when I displace it down by a distance of X, extra volume that get displaced is A square X. A square is the surface area that into the displacement down is the volume that got removed. A square X volume is displaced. That volume displaced corresponds to the extra buoyant force. So rho VG is the extra buoyant force. So the net force is the extra force that comes in earlier net force was zero anyways, okay? Or if you are very careful, you can write down the net force as rho A square X plus X naught G. This is a net buoyant force minus MG. This is equal to mass and acceleration. That way if you write then also this thing is zero. So using this equation, again it get reduced to this only. And now you compare A is equal to minus omega square X and omega will come out to be root over, there should be G somewhere, this is G. Root over rho G by sigma A. Shraddha, is it clear? Time period is two pi by omega root over sigma A by rho G. Okay? This is the way you solve this. And this is from your textbook. Unsolved problem, NCRT, the last question is this only, which we have solved. Don't think that this is advanced level or whatever it is, this is school level, okay? So I think this is what I wanted to cover today. We have four minutes. You can ask me any doubt, anything related to whatever we have done today. Please type in your doubts quickly. From anywhere, I can discuss anything. But I'll not let you go early. Tell me all of you, anything from anywhere. It cannot happen that you don't have any doubts. Okay, it is surprising, 40 students, no doubt. Energy is a cause of a situation we'll be doing next class a bit. We are talking about something we have done today. Ask doubts on that. Is that a doubt? Have you learned everything whatever we have discussed here? I'm not taking your name, but you should understand that is not a relevant question. If we have a string instead of spring, will there be SHM? There goes, the string should be elastic, okay? Then only you may have SHM, okay? The elastic string is different from spring because when you stretch it, it behaves like a spring but it can't get compressed. The tension will become zero. So string is a little tricky to deal with, okay? If you have this kind of situation, you know, something like this, one elastic string is from, let's say this side and one elastic string from this side. One string should always be extended. Then only it will be an SHM, okay? A single string like this won't create an SHM, okay? So in the next class, somebody was asking what we'll be doing, that was his doubt. So next, we'll be doing energy, all right? And we'll be talking about the potential energy, kinetic energy, and we will basically have enough time to do the force oscillation and damped oscillation as well. So next class probably we will complete the chapter itself. Fine, this is what is the plan. And I have observed that the batch previous to you from NPS Rajaji Nagar, they were doing a lot much, a lot much harder work than you guys are doing currently. Okay, and that is a fact. You can check. I think Ritu can vouch for the fact. No, because her brother was with us, but he was two years before. So someone like Archit might be knowing his brother and his brother might be knowing the other batch match. So you can talk to him, how much hard work they used to do. So I have seen that, I'm not saying completely the hard work is missing, but I'm just comparing the two batches, okay? So don't look for shortcuts. I have not seen, there are few students who are continuously not doing homework week after week. It is all going to pile up and later on, nobody can help. Even I cannot help you. And then you'll be at a sorry stage that we cannot help. So don't create that problem to yourself. Follow the lecture, follow the pace of the class. You have seen that whatever I was saying that we are just done with the curriculum just before your final exam. So you don't have time for, first I will finish the syllabus and then you sit and finish your, and it won't happen that way. You'll not get time. It is not like class 9th and 10th where if teacher finishes the curriculum then you get five months to finish yourself. That won't happen, okay? So still there is some time to change your attitude towards studies. Make sure you follow the class. It's very, very important to complete things in time for 11th and 12th, okay? It's extremely important. I cannot emphasize this enough. You don't want to make a mistake which you cannot correct later on. I have seen at least 10 years of people passing out. I'm telling you that. All right, so that's it from my side. We will meet next week. Happy holidays. Enjoy, but don't forget to do your assignments, okay? Bye for now.