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We learned in Sections 4.2 and 5.1 that a particle is in equilibrium—that is, the particle does not accelerate—in an inertial frame of reference if the vector sum of all the forces acting on the particle is zero, gF S ∙ 0. For an extended object, the equivalent statement is that the center of mass of the object has zero acceleration if the vector sum of all external forces acting on the object is zero, as discussed in Section 8.5. This is often called the first condition for equilibrium:

First condition for equilibrium: For the center of mass of an object at rest to remain at rest, the net external force on the object must be zero.

A second condition for an extended object to be in equilibrium is that the object must have no tendency to rotate. A rigid body that, in an inertial frame, is not rotating about a certain point has zero angular momentum about that point. If it is not to start rotating about that point, the rate of change of angular momentum must also be zero. From the discussion in Section 10.5, particularly Eq. (10.29), this means that the sum of torques due to all the external forces acting on the object must be zero. A rigid body in equilibrium can’t have any tendency to start rotating about any point, so the sum of external torques must be zero about any point. This is the second condition for equilibrium:

Second condition for equilibrium: For a nonrotating object to remain nonrotating the net external torque around any point on the object must be zero.