What are concurrent forces ? Explain equilibrium of particle under effect of concurrent forces.

Concurrent forces: If line of action of all given forces passes through same point then, these forces are called concurrent forces.

In mechanics when resultant force acting on a particle is zero, then particle is said to be in equilibrium. In this case particle is either stationary or moving with constant velocity.

If only one force $\vec{F}$ act on a particle, then it has accelerated motion. It cannot remain in equilibrium.

If force $\overrightarrow{\mathrm{F}}_{1}$ and $\overrightarrow{\mathrm{F}_{2}}$ act on a particle, then for equilibrium $\Sigma \overrightarrow{\mathrm{F}}=0$ means

$\overrightarrow{\mathrm{F}_{1}}+\overrightarrow{\mathrm{F}_{2}}=0$

$\therefore \overrightarrow{\mathrm{F}_{1}}=-\overrightarrow{\mathrm{F}_{2}}$

This condition is shown in figure,

If three force $\overrightarrow{\mathrm{F}_{1}}, \overrightarrow{\mathrm{F}_{2}}$ and $\overrightarrow{\mathrm{F}_{3}}$ act on a particle, then for equilibrium $\Sigma \overrightarrow{\mathrm{F}}=0$

$\overrightarrow{\mathrm{F}_{1}}+\overrightarrow{\mathrm{F}_{2}}+\overrightarrow{\mathrm{F}_{3}}=0$

$\therefore \mathrm{F}_{3}=-\left(\mathrm{F}_{1}+\mathrm{F}_{2}\right)$

This is represented in diagram shown below,

By parallelogram law of forces resultant force of $\overrightarrow{\mathrm{F}_{1}}$ and $\overrightarrow{\mathrm{F}_{2}}$ is represented by diagonal. When force $\overrightarrow{\mathrm{F}_{3}}$ equal to same magnitude is applied in opposite direction particle will be in equilibrium. By triangle of vector,

$\overrightarrow{\mathrm{PQ}}+\overrightarrow{\mathrm{QR}}+\overrightarrow{\mathrm{RP}}=0$

$\therefore \overrightarrow{\mathrm{F}}_{1}+\overrightarrow{\mathrm{F}}_{2}+\overrightarrow{\mathrm{F}}_{3}=0$

$\therefore \quad \Sigma \overrightarrow{\mathrm{F}}=0$