Can a couple be said to be dynamic equilibrium if yes does its resultant moment not contradict the definition of equilibrium?

Answer 1

No, a couple is not in equilibrium, unless there is an equal and opposite force acting against each member of the couple.

Be sure to remember that something can still be moving even if it is in equilibrium. It just means it's not accelerating.

A2 People often forget that there is a difference between a hypothetical point sized object and a typical sized object when applying the condition for equilibrium. If the forces on an object are equal and opposite it is only the centre of mass that is required to be at rest (or moving at constant velocity). So for a point sized object that means the entire object must be at rest. But for a typical sized object only the centre of mass is required to be at rest. If the applied forces are off centre (as with a couple) it is still possible for the object to rotate about its centre of mass. This is dynamic equilibrium. If one also applies the second condition for equilibrium then you can have static equilibrium (no motion of centre of mass and no rotation) if the system is initially at rest. The second condition for static equilibrium is that there be no net torque on the object. That is the off centre forces tending to rotate the object(say clockwise) must be balanced by other off centre forces which tend to rotate the object counter clockwise.