Statement of Work-energy principle:

• Change in Kinetic energy of the particle is equal to the total work done by number of forces in moving a particle through a displacement 's'.

$\sum Work\ done = Change\ in\ Kinetic\ Energy$.

• Work energy principle gives us a a relation between force, mass, velocity and displacement.

Proof:

• Consider a particle of mass 'm' at the Orgin O of Cartesian system. Let force 'F' move the particle from O to A making a displacement of OA=x. Let 'u and ''v' be the velocity of particle at point O and point A respectively.
• Force acting in the direction of displacement$= F\ cos\ \theta$
• By Newton's II law of motion,$\sum F=m\ a$

$Hence, F\cos \theta=m \dfrac {\mathrm dv} {\mathrm dt}$

$Hence, F\cos \theta=m \dfrac {\mathrm dv} {\mathrm dx} \times \dfrac {\mathrm dx} {\mathrm dt}$

$Thus,\ F\cos \theta=m.\dfrac {\mathrm dv} {\mathrm dx} \times v$

$Hence,\ F\cos \theta\ dx=m v\ {\mathrm dv} $

• Integrating we get,

$\int\limits_0^x F\cos\theta\ dx = \int\limits_u^v mv\ \mathrm dv$

$ F\ \cos\theta \bigg [x \bigg]_0^x = m \bigg [\dfrac {v^2}2 \bigg]_u^v $

$ F\ \cos\theta \ [x-0] = \dfrac {1}2m [v^2-u^2]$

$ F\ \cos\theta. \ x = \dfrac {1}2m v^2 -\dfrac {1}2mu^2$

• Hence, Work done = Change in K.E, Thus Work-Energy principle is proved.