Static Equilibrium

Static equilibrium is the state of an object that is at rest and stays at rest because all the forces and torques acting on it cancel out. Nothing accelerates and nothing rotates. A book on a table, a hanging sign, and a balanced seesaw are all in static equilibrium. It is the special case of equilibrium where the object is not moving at all, as opposed to moving at constant velocity.

An object is in static equilibrium when both of these hold:

• Net force is zero — the forces balance in every direction, so ΣF = 0 (that is, ΣFₓ = 0 and ΣFₑ = 0). This is translational equilibrium.
• Net torque is zero — the turning effects balance, so Στ = 0. This is rotational equilibrium.

If either condition fails, the object either accelerates or starts to rotate, and it is no longer in equilibrium.

The key tool for any static equilibrium problem is a free-body diagram: a sketch of the object showing every force acting on it as an arrow. Once each force is drawn and labeled, you resolve the forces into x and y components and apply the two conditions above.

1. Draw a free-body diagram and label every force.
2. Choose x and y axes and resolve each force into components.
3. Set the sum of horizontal forces to zero and the sum of vertical forces to zero.
4. Pick a pivot point and set the sum of torques to zero (choosing the pivot at an unknown force removes it from the torque equation).
5. Solve the resulting equations for the unknowns.

A 20 N sign hangs from the middle of a light horizontal rod supported by two end cables. By symmetry each cable carries half the weight, so each tension is 10 N — the vertical forces sum to zero (10 + 10 − 20 = 0) and, with the load centered, the torques about either end also cancel.

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• Physics