# Rotational Equilibrium

Consider the flywheel shown here. Any force that we apply to the flywheel will be resisted by a reaction at the supports, so the flywheel will not translate – it is fixed in position. It has translational equilibrium.

However the flywheel can still rotate, and rotation, while it is essential in a flywheel, is not acceptable in a structure. Structures must have rotational equilibrium – any forces causing rotation must be balanced by equal and opposite forces resisting rotation.

Rotation is caused when the applied force and the reaction do not line up. The larger the force, the larger the rotation effect. The larger the offset, the larger the rotation effect. Thus the rotation effect, called the applied moment, is a product of the force and its offset distance from the reaction.

An applied moment is the product of a force and the perpendicular distance from the reaction to the line of action of that force

Rotational equilibrium occurs when every applied moment acting on a structure (every force x distance) has an equal and opposite resisting applied moment

Whereas forces act along a direction, applied moments act about an axis. To ensure rotational equilibrium the sum of applied moments about any axis must be zero. This will be so if the sum of applied moments about any three mutually perpendicular axes is zero,

If the structure can be modelled in 2 dimensions (and many structures can) then all forces will act about an axis perpendicular to the plane of the structure (that is in the z direction, or out of the plane of the page). So for 2 dimensional structures, if equilibrium is to be satisfied we require

In summary the equations of equilibrium are:

- Required Reading (10 minutes):
- Hibbeler, R.C., Statics and mechanics of Materials, Section 4.1.