Mechanical equilibrium

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A body is said to be in mechanical equilibrium if the total force acting upon it is zero and the total torque acting on it is zero.[1] As the net force and torque on a body are zero, it will continue to move in a straight line with constant velocity and rotate with a constant angular velocity. If the velocity of a body in mechanical equilibrium is zero, then it will remain at rest.

If the forces acting on a body are all conservative (all forces can be expressed as the derivative of a potential), then mechanical equilibrium in one dimension can also be defined where the rate of change of the potential, V, is zero:

In three dimensions force is a vector. Mechanical equilibrium can then expressed as the gradient of the potential being equal to zero:

Types of Mechanical Equilibrium

If an object is in mechanical equilibrium and at rest, then it will remain at rest. However, if the object is displaced (moved) slightly away from equilibrium, then one of two things may happen depending on whether the equilibrium is stable or not.

If the equilibrium is stable, then the potential near the equilibrium point is like a bowl: around the centre of the bowl (the equilibrium point) the potential is greater than at the equilibrium point and the object moves back to equilibrium. This is true if the second derivative of the potential is less than 0. An example of this is simple harmonic motion and another is a ball in a bowl.

If the equilibrium is unstable, the potential is like an upside down bowl. If the object is displaced slightly, then it moves further and further away from the equilibrium point. This occurs when the second derivative in greater than 0. An example of this is a ball rolling off an upside down bowl.

If the second derivative of the potential is zero, then higher order derivatives must be examined. If more than one dimension is under consideration, it is possible for the equilibrium to be stable in one dimension but unstable in another.

References