Good morning. Today, we'll discuss about simple hybrid motion. What is simple hybrid motion? Any periodic motion, which we treat itself after equal intervals of time, is known as simple hybrid motion. Now, what is periodic motion? Periodic motion is a oscillatory motion, which prohibits itself after equal intervals of time. You may get doubt. The simple harmonic motion and the periodic motion are defined in the same way. Yes, both are almost same, but there is a distinctive difference between these two. The simple harmonic motion is a type of periodic motion, but it has to satisfy three conditions. What are those three conditions? Condition number one. There should be an inertial force which makes the body at particle moving across the mid position or equilibrium position. Condition number two. There should be an accelerates side. There should be a restoring force to accelerate the body towards the midpoint or equilibrium position. Condition number three. There shouldn't be any frictional force. If any periodic motion satisfies these three conditions, then that motion is called as simple harmonic motion. Simple harmonic motion can be a represented mathematically in two different ways. One is vectorial representation. Another one is complex number representation. Let us see how we can represent it in the general way of the vectorial representation. So, let's consider a particle of mass M is under simple harmonic motion. At any given point of time, D, the displacement of the particular simple harmonic motion can be represented with x equals 2. A is sine omega D or A past omega D. So, by these two equations, you can represent the simple harmonic motion. Now you can get it out. So, why you are using only sine and why you are using only sine omega D? Whenever you are defining mathematically, you require some mathematical functions for that. Those mathematical functions has to match with your theoretical definition. What is your theoretical definition? The motion has to be repetitive in nature and periodic in nature. Now, you list out all the mathematical functions. You have trigonometric functions. You have logarithmic functions. You have exponential functions. You have algebraic functions. Out of all these functions, which mathematical function matches with your theoretical definition? All right, these two functions. Sine and cosine. Check, sine and cosines are oscillating will be maximum on the moment that you stay excited plus 1 minus 1 or the maximum on minimum values. And they oscillate about its mean position nothing but 0. Is it? So, sine and cosines are the only functions which are repetitive in nature. Let me clarify your topic. Take the best example, e power x or log x or ln of natural logarithm of x. Can we represent our simple harmonic motion? No, you cannot. Why? They are infinite functions. See, e power x, it keeps on going, going, going. And they, it never comes back. Logithomics, it never comes back. So, they are not repetitive in nature. So, you cannot use those function to represent your simple harmonic motion. Take that, cortex, secant, cosecant. Even, you cannot use those functions in order to represent your simple harmonic motion. That is the reason why you are returning vibrations, you can see these two equations. Both are, both can be used. Now, yes, you can represent, you can mathematically define the simple harmonic motion with these equations. So, where x is the displacement, is the amplitude of the particle or system at any given point of time? Omega is the frequency and t is the time. Now, in order to analyze the vibrations completely, will the displacement elements be sufficient? No. Along with the displacement, you require velocity and acceleration. So, let's get from this displacement. You know, very little velocity is differentiation of displacement, only once you have to. So, a cos omega t, I am using this equation. Yes, I know omega t, differentiation of that, a cos omega t into omega t. So, now, I want to get the acceleration dv by dt or d square x by dt square, I can write x double dot. That equals to minus e sin omega t down omega square. I can further simplify this x double dot equals to minus omega square e sin omega t. So, I can write minus e sin omega square into x. Right? What you can conclude here? Your displacement function is here. Your velocity function is here. Your acceleration is function is here. All these three functions are important. If you are to analyze the vibrations completely, displacement, activity, velocity and the acceleration. Check the acceleration function. What you can observe? Can you conclude anything from this? Yes. You can conclude. What you can conclude? Always in simple harmonic motion, your acceleration is proportional to the displacement. Always your acceleration and the displacement are opposite in nature. Things double dot and x. But because you have negative side. Always your acceleration in simple harmonic motion is minus omega square times to the root of the displacement. These three things, you can carry out or you can conclude with the final form of the acceleration of simple harmonic motion. So, this is about vectorial representation of simple harmonic motion. You