Question

State work-energy theorem. Prove it for a variable force.

Answer 1

Work energy theorem -

It states that the work done by the net force acting on a body is equal to the changed produced in kinetic energy of the body.

Let F be the variable force

$\therefore$ Work done by this variable force , $W={\int }_{x_i}^{x_f}F\cdot dx$

where $x_i$ is the initial position

and $x_f$ is final position.

Also Kinetic energy of an object , $K=\frac{1}{2}m{v}^2$

$⟹\frac{dK}{dt}=mv\frac{dv}{dt}$

$⟹\frac{dK}{dt}=ma\frac{dx}{dt}$

$⟹\frac{dK}{dt}F\frac{dx}{dt}$

$⟹dK=Fdx$

$⟹{\int }_{K_i}^{K_f}dK={\int }_{x_i}^{x_f}F\cdot dx$

$⟹\Delta K=W$

Where $\Delta K$ is the change in kinetic energy