 So, warm welcome to the third session of this course. In this third session, we shall look at slightly more complex abstractions. In the previous session we looked at very simple abstractions, just one element at a time, the mass and correspondingly the capacitance and their describing relationships. Now, this time in the mechanical context, we shall take a mass once again, but where the mass has two aspects to it, an inertial aspect and the fact that it is in a medium where there is a force which has to overcome on account of friction or viscous resistance. In the capacitance context, we shall now take care of resistive elements and we shall now describe a slightly more complex or composite abstraction from both of these contexts and understand how solving one abstraction allows us to understand two different real life context. So, let us come down to business right away. So, let us take the mechanical system. In the mechanical system, we shall assume that we have a mass placed in a medium, let us just draw the medium around it so that if I applied a force to this mass treating the force as an agent if you like, the agent has to overcome two aspects of the mass. The force is of course a function of time. The force needs to overcome the inertial aspect of the mass which is described by m dv dt or dv t dt where vt is the mass, the velocity with which the mass moves. Of course, here I am using velocity and speed somewhat interchangeably because it is all in one direction and the other component that this force takes care of is the viscous or the resistive component, resistance, frictional component, frictional or viscous or resistive component, resistance to motion. And let us assume a very simple model for that resistive motion which is also true to some approximation in many contexts namely that that resistive behavior or that frictional behavior is proportional to the velocity with which the mass moves with a constant of proportionality given by kappa dv. So, let us keep this equation in mind and let us draw another electrical system with a capacitance and a resistance. We have a resistance and a capacitance connected in series and a voltage source placed here to excite them. So, here of course, let us assume the capacitive voltage to be vc as a function of t and the capacitance itself to have value c and a resistance in series to have the value r. We know what the current in the circuit is. The current in the circuit is c dvc t dt and the resistor voltage is of course r times i t. So, r c dvc t dt, whereupon the resistor voltage plus the capacitor voltage is equal to the source voltage. Let us call the source voltage v in t. So, the equation that we have describing this circuit is v in t is rc dvc t dt plus vc t. Now, if you like you could just divide both sides by r and we will get 1 by r v in t is c dvc t dt plus 1 by r vc t and compare this to the equation that we had for the mass and viscous of friction system. The agent for the force had two parts to which it needed to react m dvc dt dv, the inertial part and kappa 0 times vt. We see exactly similar equation here. So, here we have a composite of two elements of different kind, a so called capacitive element and a so called resistive element in both contexts. Now, you know physically the capacitive element tends to store energy and then of course, to give up energy depending on how it works and the resistive element only dissipates energy in both contexts that is true. The mass is in some sense capable of storing energy and then giving it up as required and in that case the frictional element has no capacity for storage it only dissipates wastes. So, when we understand one of these systems and bring out an abstraction to that system we have automatically dealt with the other system that is the beauty of abstraction. In fact, we can now solve this differential equation. So, you see if you took the electrical circuit which perhaps is easy to understand to many of us and if you had a constant voltage source or a battery connected here of value v0 and we assume that vc at 0 plus that means at the very beginning is 0 capacitor initially uncharged then as t tends to plus infinity or asymptotically vc t would take the same value as the source voltage and now one can understand the mass and resistive element system too. So, here if you looked at the equation that describes the mass and the viscous or frictional element something similar would happen. Once we apply a voltage of you know the voltage analogous here to an agent force which is constant in time. So, there is an agent force which is constant in time and you start pulling the mass from a 0 initial velocity or very small initial velocity if you like because you know you do not want to worry about the issues of static and dynamic viscous behavior and so on. Then ultimately the mass would need a constant velocity given by the ratio of the force to that constant frictional constant. Now, there is a beauty of solving one system and understanding both systems at once. When I solve the RC system and I understand what it does with a constant input for t greater than equal to 0 I immediately know what happens in the mass and frictional system with a constant force with almost 0 velocity at t equal to 0 in the asymptotic sense. We have now understood in this session the beauty of slightly more complex abstractions not just one element but when we interconnect elements and we solve for one interconnection we have automatically seen what happens in both contexts. This is of course we are going to do all two signals in system. We are going to build on abstractions to give us insights into many kinds of practical systems at once. And for a moment we might even forget about the practical context from which that abstraction was derived. In fact, for a long time in the course we shall forget about the context from which those abstractions are derived and we shall focus on the conclusions that we would draw from those abstractions and then finally translate those conclusions back into the real world. That is why we are going to learn many beautiful ideas that can be derived living at the level of abstraction on its own. We shall see more in the next session.