\n In our daily life, we refer to various activities as work. For example-when we sit and read a book, we say we\r\n are doing work. Try pushing a wall, after few minutes you will be tired and you will say that you have done\r\n work. But scientifically, all these activities are not considered as work.\n
\r\n\n Work is said to be done when the force applied on an object makes the object move (i.e., produces a displacement\r\n in it). In short, work is said to be done only when an object moves under the influence of the force applied on it.\n
\r\n\n The amount of work done is equal to the product of the force applied and displacement produced in the body in\r\n the direction of the force applied. It is equal to the dot product of the force applied and the\r\n displacement. It is given as,\n
\r\n $$W=\\vec{F} \\cdot \\vec{S}= FS \\cos{\\theta} $$\r\n\n where \\(F\\) is the force applied on the object, \\(S\\) is the displacement, and \\(\\theta\\) is the angle between\r\n them. Work is a scalar quantity.\n
\r\n\n Work done on a body can be zero, negative or positive depending on the angle between the direction of force\r\n applied and displacement of the body. For example, when a stone tied to a string whirls around the string,\r\n no work is said to be done even though tension force in the string applies a centripetal force on the stone.\r\n It is because the displacement of the stone at all instants is normal to the direction of the tension force\r\n applied on the stone.\n
\r\n\n S.I unit of work is Newton-meter or joule.\r\n One joule is the work done by a force of 1 newton in displacing the body through a distance of 1 meter in\r\n its own direction.\n
\r\n\n Energy is defined as the capacity of a body to do work. If a body can do work, it is said to possess energy.\r\n Its S.I unit is joule. Whenever work is done, energy transfer takes place.\n
\r\n\n There are different forms of energy, for example solar energy, heat energy, light energy, mechanical energy,\r\n sound energy, magnetic energy etc. But primarily, there are two forms of energy, kinetic energy and potential\r\n energy.\n
\r\n\n The energy possessed by a body due to its state of motion is called kinetic energy. If a body is moving,\r\n it possesses kinetic energy. For a body of mass m moving with a velocity \\(v\\), its kinetic energy \\(K\\) is\r\n given as,\n
\r\n $$K=\\frac{1}{2}mv^2$$\r\n\n The energy possessed by a body because of its position, arrangement or state is known as potential energy.\r\n Potential energy can convert to kinetic energy. Potential energy possessed by a body due to the\r\n gravitational force applied on it by the earth is known as gravitational potential energy. It is the most\r\n common form of potential energy that we will study and is equal to \\(m \\times g \\times h\\), where \\(m\\) is\r\n the mass of the\r\n body, \\(g\\) is the acceleration due to gravity and \\(h\\) is the height of the body.\n
\r\n\n We have studied that work done on a body makes the body move and its energy changes. This change in energy\r\n of the body is the change in its kinetic energy.\r\n The work-energy theorem states that the work done by the force applied on a body is equal to the change in\r\n its kinetic energy.\r\n
\r\n $$W=\\Delta K =K_f - K_i$$\r\n\n Let us derive the work-energy theorem.\r\n Consider a body of mass \\(m\\) moving with an initial velocity \\(u\\). A force \\(F\\) is applied on the body which\r\n produces an acceleration \\(a\\) and displaces it through a distance \\(s\\). The final velocity of\r\n the body is \\(v\\). Let \\(W\\) be the work done on the body throughout the entire process.\r\n
\r\n $$\r\n \\begin{equation} \\label{eq2}\r\n \\begin{split}\r\n \\text{Work}&= \\text{Force} \\times \\text{displacement}\r\n =F \\times s \\;\\;\\;\\;\\;\\; -(i)\r\n\r\n \\end{split}\r\n \\end{equation}\r\n $$\r\n\r\n\n By the third equation of motion, we know \\(s=\\frac{v^2-u^2}{2a}\\). Therefore, replacing \\(s\\) in equation (i), we\r\n get\n
\r\n $$\r\n \\begin{equation} \\label{eq3}\r\n \\begin{split}\r\n W&= F \\times \\left( \\frac {v^2-u^2}{2a} \\right)\r\n \\\\&=ma \\times \\left( \\frac{v^2-u^2}{2a} \\right) \\;\\;\\;\\;\\;\\;\\;\\; \\because F=ma\r\n \\\\&=\\frac{m(v^2-u^2)}{2}\r\n\r\n \\\\&=\\frac{1}{2}mv^2-\\frac{1}{2}mu^2\r\n \\\\&=K_f-K_i\r\n \\end{split}\r\n \\end{equation}\r\n $$\r\nThus, we have derived that work done on a body is equal to the change in its kinetic energy.
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