What is torque? Explain the torque acting on a particle.

(N/A) The role of torque in rotational motion is similar to the role of force in translational motion.
Consider a force $\vec{F}$ acting on a particle $P$ having a position vector $\vec{r}$ with respect to the origin $O$. The angle between $\vec{r}$ and $\vec{F}$ is $\theta$. The vector product of $\vec{r}$ and $\vec{F}$ is defined as the torque $\vec{\tau}$ acting on the particle with respect to the origin $O$.
$\therefore \vec{\tau} = \vec{r} \times \vec{F}$
The magnitude of torque is given by:
$\tau = r F \sin \theta$
where $|\vec{r}| = r$ and $|\vec{F}| = F$.
Since $\tau = r F \sin \theta$,we can rewrite this as:
$\tau = (r \sin \theta) F = r_{\perp} F$
where $r_{\perp} = r \sin \theta$ is the perpendicular distance of the line of action of the force from the origin.
Alternatively:
$\tau = r (F \sin \theta) = r F_{\perp}$
where $F_{\perp} = F \sin \theta$ is the component of the force perpendicular to the position vector.
Thus,torque is the moment of force with respect to point $O$.