Stability

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For an object to be in mechanical equilibrium, the net external force and the net external torque acting on the object have to be zero.

No net external force implies that the center of mass of the object is at rest or moving with constant velocity.  No net external torque implies that the object either does not rotate or rotates with constant angular velocity.  For an object in mechanical equilibrium there exists some inertial reference frame in which the object's center of mass is at rest.  If in this frame the object also does not rotate, it is in static mechanical equilibrium.

Equilibrium does not guaranty stability.

Example:  A ball and a bowl
 
The ball is in stable equilibrium at the
bottom of a bowl.
The ball is in unstable equilibrium at the
top of a bowl.

 

Objects are in stable mechanical equilibrium if their potential energy increases if they are slightly disturbed.