5.1:
Conditions of Equilibrium
A rigid body does not undergo any deformation under the influence of external forces.
Consider a merry-go-round subjected to an external force F at a perpendicular radial distance r, which results in the moment of a force about point O.
If multiple forces act on the merry-go-round, the equilibrium condition is obtained when the sum of all external forces acting on it is zero.
The other condition of equilibrium states that the sum of all moments about point O must equal zero.
When the moment is measured at point A, it can be expressed as the sum of the moment at point O and the cross product of perpendicular distance and the applied force.
These conditions ensure that the body does not undergo any translational or rotational motion in static equilibrium.
Only external forces are considered here since the internal forces within the rigid body have the same magnitude and act in opposite directions, canceling each other due to Newton's third law of motion.
5.1:
Conditions of Equilibrium
Equilibrium refers to a state where a rigid body is not subjected to any translational or rotational motion. This state is achieved when the force and couple acting on a rigid body equal zero. When the system of external forces results in a net effect equivalent to zero, the rigid body is considered to be in equilibrium.
Internal forces are not considered for conditions of equilibrium because they occur in equal and opposite pairs within the body, effectively canceling each other. As a result, these internal forces do not contribute to the overall motion or change in motion of the rigid body. Equilibrium is determined by the balance of external forces and moments acting on the body, as they are responsible for causing translational or rotational motion.
The necessary and sufficient conditions for equilibrium can be determined by setting the resultant force and moment equal to zero. Additionally, the vector equations can be replaced with six scalar equations by resolving each force and moment into its rectangular components. This implies that the components of the external forces in the x, y, and z directions are balanced, and the moments of the external forces about the x, y, and z axes are also balanced. If a rigid body is in equilibrium, the system of external forces imparts no translational or rotational motion to the body.
To write the equilibrium equations for a rigid body, first, identify all the forces acting on the body and then draw the corresponding free-body diagram. Besides the forces applied to a structure, consider the reactions exerted on the structure by its supports. Correctly identifying each reaction can help solve the unknown forces and reactions.
Knowledge of equilibrium conditions is crucial in analyzing problems related to rigid bodies. By determining the unknown forces and reactions and using the necessary and sufficient conditions for equilibrium, it is possible to predict the behavior of rigid bodies.
Suggested Reading
- Beer, F.P.; Johnston, E.R.; Mazurek, D.F; Cromwell, P.J. and Self, B.P. (2019). Vector Mechanics for Engineers ‒ Statics and Dynamics. New York: McGraw-Hill. pp 170