Question

State and derive work energy theorem.

Answer 1

Work energy theorem states that the change in kinetic energy of an object

is equal to the net-work done on it by the net force.

Let us suppose that a body is initially at rest and a force.

$\overrightarrow{F}$

is applied on the body to displace it through

$d\overrightarrow{s}$

along the direction of the force. Then, a small amount of work done is given by

dw = $\overrightarrow{F}.d\overrightarrow{s}$ = Fds

Also, according to Newton's second law of motion, we have

F = ma

where a is acceleration produced (in the direction of force) on applying the force. Therefore,

dw = Mada = $M\frac{dv}{dt}ds$

Now, work done by the force in order to increase its velocity from u (initial velocity) to v (final velocity) is given by

W = ${\int }_u^{v}Mvdv$ = $M{\int }_u^{v}vdv$

= $M|\frac{v^2}{2}{|}_u^v$

W = $\frac{1}{2}M{v}^2-\frac{1}{2}M{u}^2$

Hence, work done on a body by a force is equal to the change in its kinetic energy.