An object in rotational equilibrium has no net external torque:
sum of all external torques = 0
Does it mean "rotational equilibrium" may mean that the object is not rotating … or it may mean that the object is rotating with constant angular velocity? But when it is rotating ,is it still an equilibrium ?
Rotational equilibrium: $\sum \tau_C=0$
Note that for rotational equilibrium of a rigid body, the net torque about its centre of mass must be zero.
In this quote from a book it says that ror rotational equilibrium, torque about the body's Center of Mass must be zero. But why only about the center of mass? About any other point also it is in equilibrium. It is confusing.
Best Answer
Equilibrium can mean different things depending on the context. For example think of a container of gas with a permeable barrier dividing the container in half. At equilibrium you would expect an equal concentration of gas molecules on either side of the partition even though there is still going to be a transfer of gas molecules between each side of the container. The net flow is $0$ though, so we say the system is in equilibrium.
In the context of torques, I would all we need for equilibrium is that the net torque is $0$. Whether or not the system in question has a net rotation depends on your "initial conditions". Typically then you see in many statics scenarios that the system in question is also at rest, but you wouldn't approach the problem any differently if there was rotation (unless the motion changes torques of the system, but then it turns into a dynamics problem instead of a statics problem).
I think this is the distinction between "rotational equilibrium" and more of a complete "mechanical equilibrium". For example, if there is a single force acting on the center of mass of our system, then of course there is no torque about the center of mass of the system and there will be no rotation about the center of mass. However a single force cannot keep a system in mechanical equilibrium, as there will be a net force. But if we pick any other point not along the direction of the force we will find a net torque about that point.
For complete mechanical equilibrium we can choose any point about which to find the torque and it must be $0$. But for just rotational equilibrium we just need to consider torques about the center of mass.