**Introduction**

**Work done** is defined as the product of the distance traveled by a body when force is applied in any direction except perpendicular to force.

**Work done by a constant force** is defined as the displacement produced by a constant force when it acts on a body.

If θ is the angle between F and s as shown in the figure (B), then from equation(I) we have

W = F. s.cosθ …………(II)

If displacement and the force is in the same direction as shown in figure(I), then θ = 0, cosθ = 1,

W = F. s. 1

=F. s

**Units of works**

1J = 1N * 1m = 10^{5} dyne * 100cm = 10^{7} erg

**Work done by Variable force**

Let a variable force is acting on a body to displace it from A to B in a fixed direction. We can consider the entire displacement from A to B is made up of a large number of infinitesimal displacements. One such displacement is shown in the figure from P to Q. As the displacement PQ = dx is infinitesimally small, we consider that along with the displacement, force is constant in magnitudes as well as in direction.

Small amount of work done in moving a body from P to Q is

dW = F * dx = (PS) (PQ) = area of strip PQRS

Total work done in moving body from A to B is given by,

W = ∑dW = ∑F. dx

If the displacements are allowed to approach zero, then the number f terms in the sum increases without limit. And sum approaches a definite value equal to the area under the curve CD as shown in the figure.

Hence, we can write

W = lim dx -> 0 ∑f (dx)

= ^{xb}ʃ_{xa} F dx

=^{xb}ʃ_{xa} area of strip PQRS

= total area under the curve between F and x-axis from x = x_{A} to x= x_{B}

W = Area ABCDA

Hence work done by a variable force is numerically equal to the area under the force curve and the displacements.

**Energy** is defined as the capacity to do work.

**Kinetic Energy** is the energy possessed by a body due to its motion.

**Expression for Kinetic Energy**

Let us consider a block of mass lying on a smooth horizontal surface as shown in figure. Let a constant force F be applied on it such that, after traveling a distance s, its velocity becomes v.

If a be the acceleration of the body, then

V^{2} = u^{2} + 2as = 0 + 2as

Or, v^{2} = 2as

as initial velocity, u = 0,

So, as = v^{2}/ 2 …………… (i)

As workdone on the body = force * distance, then

W = F * s

= m. a. s ………………. (ii)

From equations (i) and (ii), we have

W = m * v^{2}/2 [as = v^{2}/ 2]

= ½ mv2 is the expression for Kinetic Energy of a body

**Potential Energy** is the energy possessed by a body due to its position or configuration.

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