Work - Energy Principle:

• Work : The work done by a constant force $F$ is the product of the component of force in the direction of displacement and the magnitude of displacement. $$ \mathrm{du}=\int_{0}^{s}[F \cdot \cos \theta] d s $$ Work done $=\Sigma$ [ Force in the direction of motion] x displacement.

Kinetic Energy:

It is the energy possessed by a particle by virtue of its motion. If the particle of mass ' $\mathrm{m}$ ' is moving with a velocity ' $\mathrm{v}$ ' then its kinetic energy is K.E $=1 / 2 \mathrm{mv}^{2} \mathrm{Nm}$. i.e Joule in SI units.

From Newton's II Law $\mathrm{F}=$ ma $\quad$ but $\mathrm{a}=\frac{d v}{d t}=\frac{d x}{d t} \frac{d v}{d x}=\mathrm{v} \frac{d v}{d x}$ $$ \begin{gathered} \mathrm{F}=\mathrm{m} \mathrm{v} \frac{d v}{d x} \\ \mathrm{~F} d x=\mathrm{m} \mathrm{v} d v \\ \int_{x 1}^{x 2} \mathrm{~F} d x=\mathrm{m} \int_{v 1}^{v 2} v d v \\ U_{1-2}=1 / 2 \mathrm{mv}_{2}^{2}-1 / 2 \mathrm{mv}_{1}^{2} \\ U_{1-2}=\mathrm{K} . \mathrm{E}_{2}-\mathrm{KE}_{1} \\ K E_{1}+U_{1-2}=K E_{2} \end{gathered} $$

Initial KE + work done by all force = Final KE The above equation is called the work-energy principle.