If we have to define equilibrium the most simple definition would be it is a point where the net external force as well as torque acting on the body about COM or any other point is zero. But to be more specific for a rigid body equilibrium means both rotational as well as translational equilibrium. For example consider the following situation:
In this case the body is in translational equilibrium but the two forces form a couple hence it is not in rotational equilibrium. So suppose for a rod kept on a table minimum number coplanar forces (having different line of actions) that must act so that body remains in equilibrium? So if we analyze with one force it can’t be either translational or rotational equilibrium, with two force translational is possible but not rotational but with three we can achieve both.
Equilibrium is classified as Dynamic or Static equilibrium. The next question is what static equilibrium is and how it is different from dynamic? So if the body is in equilibrium but continues to move with uniform velocity it is known as dynamic equilibrium. For example, a ball moving with uniform velocity. On the other hand, if the body is in equilibrium while being at rest it is termed as static equilibrium.
Equilibrium is also classified as stable, unstable and neutral. Let’s see what these mean. Stable equilibrium is one in which if the body is displaced from its equilibrium position then it tends to move towards that equilibrium point. For example a ball kept at the bottom of a hemisphere.While in case of unstable equilibrium, if it is displaced from that point the body tends to move away from that point. Consider a ball kept at the top of a sphere. If we slide it, the ball tends to roll away from the topmost point. Similarly in neutral equilibrium the body neither moves towards nor away from the equilibrium point. For example displace a ball kept on horizontal surface slightly.
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