The work done by a force acting on the body during its displacement is equal to the change in the kinetic energy of the body during that displacement.

*Principle of work and energy (work - energy theorem) Statement*

*The work done by a force acting on the body during its displacement is equal to the change in the kinetic energy of the body during that displacement.*

*Proof*

Let us consider a body of mass *m* acted upon by a force *F* and moving with a velocity *v* along a path as shown in Fig.. At any instant, let P be the position of the body from the origin O. Let Î¸ be the angle made by the direction of the force with the tangential line drawn at P.

The force F can be resolved into two rectangular components :

(i) Ft = F cos Î¸ , tangentially and

(ii) Fn = F sin Î¸ , normally at P.

But Ft = mat ...(1)

where at is the acceleration of the body in the tangential direction

âˆ´ F cos Î¸ = mat ...(2)

But at = dv/dt ...(3)

âˆ´ substituting equation (3) in (2),

F cos Î¸ = m dv/dt = m dv/ds .ds/dt ...(4)

F cosÎ¸ ds = mv dv ...(5)

where ds is the small displacement.

Let v1 and v2 be the velocities of the body at the positions 1 and 2 and the corresponding distances be s1 and s2.

Integrating the equation (5),

*Therefore work done*

*= final kinetic energy âˆ' initial kinetic energy*

*= change in kinetic energy*

*This is known as Work-energy theorem.*

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