 Hello everyone, I am Swathi Ghadge, Assistant Professor, Department of Civil Engineering, Walsh and Institute of Technology, Solapur. Topic for today's session is work energy method. Learning outcome of this session is, at the end of this session, learner will be able to determine work done or energy, work energy principle, initial velocity, final velocity, displacement, acceleration and time. In this session, the numerical on initial velocity, final velocity or displacement and acceleration not covered, but by using the work energy principle, student can determine this other parameter. First, see what is work done? Work done is the product of force and the distance, more in a direction of the force. For example, at point A, the several forces are acting P1, P2, P3, P4. As a result, A displays from its old position to the new position A dash by the distance. So, we will see one by one, what is the work done by force P1, what is the work done by force P2. P1, P1 you can see here, P1 is in the horizontal direction and it is in the direction of displacement. So, for the P1 force, displacement is P1 into S. Now, see the P2 force. P2 force is not in x direction, it is inclined, let us say it is inclined at angle theta with horizontal. So, for the P2, horizontal component is P2 cos theta and vertical component will be P2 sin theta. So, P2 sin theta will not contribute in a work done and P2 cos theta will contribute because it is in the direction of displacement. So, for the P2, work done by force P2 is P2 cos theta into S. Now, see for the P3 force, P3 force is not in x direction or not even its component is in x direction. So, P3, it is not at all contribute in the displacement. So, for the P3, work done is 0. Now, see P4, P4 is in x direction, but exactly in opposite direction. So, the work done by force P4 is minus P4 into S. Now, see by energy. What is energy? It is the capacity to do work. It may be classified into potential energy and kinetic energy. Potential energy is the capacity to do work due to position of the body. So, suppose the body of weight is W held at a height h, position potential energy W into h. So, this is W into h is the potential energy and kinetic energy, it is the capacity to do work due to the motion of a body. Now, see diagrammatically how the body will possess a kinetic energy. So, on the surface there is a block or there is a body of weight W and it is moving in a, suppose it is in a motion and suddenly the brake apply. So, obviously brake apply will be in opposite direction, but still it will move with some initial velocity, let it be U and as the force is acting in opposite direction of motion. So, acceleration will act in a direction of force that is in the opposite direction of motion. Even after applying a brake, the body will still move for some distance, let the distance will be S. So, that will be the final position of the W. So, here let us understand first, here there is a body of weight W, it is in motion. So, there is some energy store in that body, suddenly it is suddenly the brake apply in the opposite direction of motion, but still the body will cover some distance with the initial velocity U. So, in the direction of force acceleration will develop and after some time the body will stop, after covering some distance the body will stop. So, let us at that point the velocity will become 0. So, when the brake apply at that point the initial velocity will be U and when the body will stop that point velocity will be final velocity will be 0. Then according to D'Alembert's, apply a force that is called as the inertia force M A is exactly opposite in the direction of motion. So, body is moved from its old position to the new position, so this is the direction of motion, in the opposite direction of motion apply a force M A. Now, we can apply a equilibrium condition that is called as the dynamic equilibrium condition for this body. So, here two forces are acting, so according to the equilibrium condition that two forces must be equal. So, F is equal to minus W by G into A. So, by using linear motion equation find out A, so we have final velocity is 0, initial velocity U put in a linear motion equation V square minus U square is equal to twice AS, V is 0 and find out A. So, A is equal to we will get A minus U square upon 2S. So, that calculated A put in a equation 1, so we will get F is equal to W U square upon twice GS. So, this is the force acting when we apply a brake. So, the work done is force into distance. So, force into distance, so that S will multiply by force. So, the remaining term is the work done that is W U square upon 2G. So, like this W U square upon 2G it is the work done. Now, see you just compare that formula with the kinetic energy formula half MV square. Kinetic energy formula is half MV square, so it is similar to the kinetic energy. So, we can say that work done is nothing but the kinetic energy. So, here kinetic energy is half W by G is the MV square. Now, you pause the video here, we discussed earlier work done and the energy. Now, you answer the question what is the unit of work done or energy? Your options are Newton into meter, joule, Newton into mm and all of this. The correct answer is all of this because work done formula is force into distance. So, Newton it is the unit of force and meter is the unit of distance. Same here third for third option Newton and mm are the distance of force and distance and the joule it is the relation of Newton meter and joule is 1 Newton meter is equal to 1 joule. So, joule it is also a unit of work done or the energy. So, correct option is all of this. Now, we will derive your work energy equation. So, for that for the explanation purpose let us consider particle A and the several forces are acting on this particle including its self-fed. So, self-fed of this particle is W acting vertically downward P1, P2, P3 and P4 some forces randomly acting on that body. Now, as a result A move from its old position to the new position let it be A dash. So, it is move in the horizontal direction it means that resultant force acting a horizontal direction by distance S. So, here resultant is we can say that summation of f x because A move in the x direction. So, resultant will be the summation of horizontal forces or summation of forces along the x axis. Here weight it is the body having the weight W. So, R is equal to ma. Now, multiply both side by ds, so R into ds is equal to W by g into A into ds as we know that dv by dt is acceleration. Now, it can be written as A is equal to dv by ds into ds by dt also. So, A is equal to v into dv by ds. Now, you put that calculated A in this previous equation we will get R into ds is equal to W by g v into dv. So, our derive equation is R into ds W by g v into dv. Now, you integrate both side for the motion from A to A dash because body move from A to A dash, it is the work done. So, we will integrate it. So, R into ds integration lower limit is 0 and because initially distance 0 and finally distance is s. Now, here W by g v into dv, so we will we are integrating it. So, for we know that initial velocity is u and final velocity is v and integration of R ds is R s, W by g is the constant term outside of the integration and v dv integration of v dv is v square upon 2. So, its upper limit is v and lower limit is u. So, after putting upper limit and lower limit will get R s is equal to W by g v square minus u square. So, R s is equal to half W by g in bracket v square minus u square, it is the work energy equation. Now, here there is a velocity in the form of v square minus u square. So, we know that final kinetic energy is half W by g v square and initial kinetic energy is half W by g u square. So, work energy principle is final kinetic energy minus initial kinetic energy. So, you remember the equation highlighted in a red color, this is the work energy equation R s is equal to half W by g v square minus u square. These are my references, thank you very much.