Write the conditions for the equilibrium of a rigid body.

$A$ rigid body is said to be in mechanical equilibrium if both its linear momentum and angular momentum do not change with time,meaning the body has neither linear acceleration nor angular acceleration.
$(i)$ Translational equilibrium:
If the vector sum of all forces acting on the rigid body is zero,then the body is in translational equilibrium.
$\sum_{i=1}^{n} \overrightarrow{F}_{i} = 0$
This implies that the total linear momentum of the body remains constant.
$(ii)$ Rotational equilibrium:
If the vector sum of all torques acting on the rigid body about any point is zero,then the body is in rotational equilibrium.
$\sum_{i=1}^{n} \overrightarrow{\tau}_{i} = 0$
This implies that the total angular momentum of the body remains constant.
These vector equations can be resolved into scalar components:
For translational equilibrium: $\sum F_{ix} = 0, \sum F_{iy} = 0, \sum F_{iz} = 0$.
For rotational equilibrium: $\sum \tau_{ix} = 0, \sum \tau_{iy} = 0, \sum \tau_{iz} = 0$.