Equilibrium

Type of physical science: Equilibrium, Newton's laws of motion, Inertia, Momentum, Vectors, Classical physics

Field of study: Mechanics

Equilibrium is a condition in which a body is at rest or, if it was previously in motion, moves with constant velocity along a straight line.

Overview

Sir Isaac Newton's first law of motion states that if an object is at rest, it will remain at rest, or if it is in motion, it will remain in motion at a constant velocity and in a straight line, unless some unbalanced force changes its condition. This unchanged condition of the object is called "equilibrium."

In other words, if a single, unopposed force acts on a body, it causes the body to move. If the force acts through the center of gravity of the body, the movement takes place in a straight line along the line of action of the force. This is called "translation." If, however, the force acts on the body, but not through its center of gravity, it not only causes the body to move but also causes the body to rotate. In either of these states, the body is not in equilibrium. Yet consider two forces acting on the body at the same time. If they are both equal in magnitude, act along the same straight line, but are in opposite directions, the body then cannot move, and equilibrium is maintained. The forces cause internal stress and strain within the body, but no movement. It can thus be seen that equilibrium exists when there is balance. An unbalanced system of forces upsets the equilibrium of the body, causing either movement or rotation along or about three orthogonal, coordinate axes, typically termed the X, Y, and Z axes.

As noted, a single force acting on the body can never keep it in equilibrium. It is easy, however, to see that more than one force can act on the body at the same time. A simple example is the case of gravity loads and wind loads acting on a tall building. In this case, the two forces act on the body at an angle to each other. How, then, could equilibrium be maintained? To answer this, it is first necessary to determine the outcome of the original force system. This can be done graphically, by completing the parallelogram of forces and determining the diagonal of the parallelogram. The diagonal, or "resultant," is the direction in which the body will actually move under the action of the two forces. The same diagonal can also be obtained by placing one force at the tip of the other. The resultant is now the third side of the triangle. If equilibrium is to be maintained, an equilibrant has to be applied to the body. This is a force exactly equal to the resultant, also acting along the diagonal of the parallelogram, but in the opposite direction. The method can be expanded to any number of coplanar, concurrent forces acting on the body, by completing the polygon of forces (or repetitively completing the triangles of forces). The resultant then is the last side of the polygon; the equilibrant is of the same magnitude of force, but in the opposite direction. When the body is in equilibrium, the polygon of forces is said to be "closed."

If two forces are not concurrent but parallel to each other, are equal in magnitude, and act in opposite directions, the body is acted upon by a couple. Since the forces are equal and opposite, there is no translation. The couple, though, rotates the body about the axis of the couple. The magnitude of the couple is determined by its moment, which is equal to one of the forces times the distance between them. In order to maintain equilibrium, therefore, the equilibrant would be a couple of exactly the same moment, but of opposite sense. A clockwise couple, for example, would need a counterclockwise equilibrant.

In the most general case, therefore, with any number of noncoplanar forces and moments acting on the body in three-dimensional space, the entire system can be reduced to resultants of forces along, and moments about, the three coordinate axes. When it can be categorically stated that the sum of all the forces about each of these three axes, together with the sum of all the moments about each of these three axes, is individually equal to zero, then the body will be in equilibrium.

Consider, once again, the forces applied to a tall building, such as vertical gravity forces caused by its own dead load and the weight of its occupants, together with horizontal wind forces acting on the height of the building. How, then, is equilibrium maintained? Newton's third law of motion states that for every action there is an equal and opposite reaction. It is this reaction that functions as the equilibrant and keeps the body in equilibrium. Consider the simple case of a rope, one end of which is attached to a wall and is being pulled by a person at the other end. The force is applied at only one end, but the rope is in equilibrium. The reason is that the wall generates a reaction that is exactly equal and opposite to the force that is being applied. If the applied force is reduced, the reaction decreases instantly; if the applied force is released, the reaction disappears. In the case of the tall building, the reactions of the soil in which the tall building is anchored provide the vertical and horizontal equilibrants that prevent the building from settling into the soil or sliding on it, thereby maintaining equilibrium.

These reactions can be, essentially, of three types. If the end of the member is supported so that it can slide across the face of the base support, such as in a roller-bearing support, then the force reaction is provided only at right angles to the face of the support. If, however, the member cannot move but is permitted rotation at the support, it is a pinned-end support. In such a support, force reactions can be developed, but no moment reactions develop that could prevent rotation of the member. Finally, if the member is anchored in such a way that the support can provide force and moment reactions about all three axes, then the support is a fully fixed end. Hybrid-end conditions such as partially fixed or partially pinned-end conditions are also possible. It is these reactions at the anchorage, whether for a tree, a tall building, or a bridge, that provide the force and moment equilibrants necessary to maintain equilibrium of the structure.

Applications

It would be impossible to imagine life without conditions of equilibrium. From the simple act of walking to the structure of a tall building, it is only the state of equilibrium that permits safety. In the case of walking, the most common reaction or equilibrant is friction. This is the force that develops between two bodies in contact with each other that prevents sliding along the common surface. The coefficient of friction is the ratio of the force developed along the surface as a proportion of the force applied between the bodies at the surface. As an example, the coefficient of friction of a rubber tire on asphalt is about 0.8. In comparison, the sliding coefficient of friction of steel on ice is about 0.014. It is easy to see that an automobile would have difficulty rolling down the street without the application of substantial power, because a large frictional reaction can be developed to prevent the wheels from spinning, whereas an ice skater, with steel skates, will slide easily without much friction on the ice. In both cases, however, the effect of wind resistance is also a reaction being developed on the body.

Equilibrium has many applications in the analysis and design of structural systems, because it assumes that an object is at rest. This implies that it is safe and failure has not occurred. Given such a condition, equilibrium permits a designer to determine the forces and moments that must exist, internally, within the structural system under the applied loads in order to maintain equilibrium. If equilibrium exists, then the resultant of all forces must be zero. This fact determines the value and directions of the equilibrants.

Consider a suspension bridge. The designer can determine the force in the cables under the applied loads on the bridge structure. This load will include the estimated weight of the bridge itself, together with the load from the lanes of traffic. The impact forces created by heavy trucks rolling down the bridge will also have been taken into account. If the bridge is to be in equilibrium, then the force in the cables has to be resisted by an equilibrant reaction that has to be provided by the anchorage of the cables in the surrounding rock. This equivalency gives the precise nature of the forces that the anchorage must provide to the bridge for safe design.

An excellent example of rotational equilibrium is the child's teeter-totter or see-saw in a playground. If a heavy child is seated close to the fulcrum, or point of rotation, the moment about the fulcrum can be balanced by a lighter child who is seated much farther away from the fulcrum. The application of a small upward force by the child at one end will enable the plank to swing upward until the feet of the child at the other end reach the ground and provide a reaction to oppose it.

The application of equilibrium to the case of moving objects, such as cars, trains, or planes, is equally important. Consider an automobile in motion. It is subject to its own vertical gravity loads, which include the weight of its occupants and the luggage in the trunk. It is acted upon by the driving forces at the wheels, and this motion is resisted both by the friction at the road surface and by wind pressure. Under all these forces, if the automobile is moving at a constant velocity in a straight line, it is in equilibrium. If the driver presses on the gas pedal, however, the automobile accelerates, and equilibrium no longer exists. The acceleration will occur until the increased wind resistance and increased road friction equal the increase in the force that is being applied to the automobile, at which time the car will return to a constant velocity and, hence, to a state of equilibrium.

Finally, it is loss of equilibrium that results in failure, sometimes with disastrous consequences. A good example is foundation failure, which occurs in buildings when equilibrium cannot be maintained. If the soil is poor, with a low safe-bearing capacity, or if an incorrect choice of the type of foundation has been made, large forces (actions) are imposed on the soil. When the equilibrant reactions are not forthcoming, the foundation caves in, and failure occurs. In the case of cantilever-retaining walls, for example, sliding of the wall, caused by the horizontal pressure of the earth, has to be resisted by friction at the base. The heel of the wall is, therefore, kept long--more than two-thirds the length of the base--so that the weight of retained earth on the heel provides large vertical forces that produce correspondingly large frictional equilibrants. The weight of the earth on the heel also prevents the wall from overturning, thereby maintaining equilibrium.

Context

Equilibrium has been intuitively understood by nature for many hundreds of millions of years, ever since the start of biological evolution. The anchorage systems developed by roots, such as buttress roots or stilt roots, to anchor tall trees are excellent examples of nature's attempts to maintain equilibrium for the safety of the organism. Anthropological humankind also understood the importance of maintaining equilibrium for the safety of shelters. Ancient humans also understood the need for a large base to prevent the overturning of structures, such as the pyramids. Even Galileo Galilei investigated the equilibrium of a cantilever. It was Isaac Newton who, with his laws of motion, set the framework for a more classical study of equilibrium. This study was prompted by the need to develop classical methods of analysis that would permit engineering design, as opposed to the intuitive design of structural systems. The foremost need was for the development of towers. City-states and governments had reached the conclusion that vertical height gave greater forewarning of attack, and the development of towers was a prime military objective. Unfortunately, this was not coupled with an understanding of the nature of soil; as a result, many of these towers leaned out of plumb. The study of dynamic equilibrium, with bodies in motion rather than at rest, took on added impetus with the Industrial Revolution and the development of the steam engine. This process continued with advances in the design of automobiles, airplanes, and space ships. Yet the principles on which equilibrium is based can never change. This is because they are, quite simply, based on fundamental laws of physics, and regardless of the applications to which the principles may be put, these laws are invariant and immutable. In an ever-changing world, it is comforting to know, perhaps, that these laws that govern the equilibrium of bodies, at rest or in motion, will forever be the same.

Principal terms

COUPLE: A system of two equal and opposite parallel forces, spaced a finite distance apart

EQUILIBRANT: A single force or couple that when applied to the system causes the body to be in equilibrium

FIXED END: The end of a member that is not permitted either sliding movement or rotation

FORCE: An action or reaction that attempts to cause the body to move along its own line of action

FRICTION: A force that arises between two bodies in contact, resisting the movement between them

PINNED END: The end of a member that is permitted rotation about one or more axes

RESULTANT: A single force or couple that can represent the applied forces and couples in their action on the body

ROLLER END: The end of a member that is permitted sliding movement in one or more directions

Bibliography

Benjamin, B. S. Structural Evolution: An Illustrated History. Lawrence, Kans.: A. B. Literary House, 1990. A well-written, heavily illustrated book about structural systems in nature and in architecture. Includes discussion of the equilibrium of structures. Nonmathematical and written for a general audience; makes some interesting links between nature and architecture.

Billington, David P. The Tower and the Bridge. New York: Basic Books, 1983. Deals with the design of large structures with great vertical heights or long horizontal spans. An excellent introduction to the equilibrium of these two types of structures.

Cowan, H. J. The Master Builders. New York: John Wiley & Sons, 1977. An excellent, largely nonmathematical work that researches structural developments in architecture, ancient, medieval, and modern. Principles of equilibrium are examined.

Feininger, Andreas. The Anatomy of Nature. New York: Crown, 1956. An excellent photographic journey through the various structural systems displayed by nature. While there is no overt reference to equilibrium, the underlying principles of equilibrium used by nature for the design of these systems are obvious.

Frisch, Von K., and O. Frisch. Animal Architecture. New York: Harcourt Brace Jovanovich, 1974. Written by a Nobel Prize winner. Remarkable for its study of the way in which insects use good structural principles, including equilibrium, for the design of their structures.

Salvadori, Mario. Why Buildings Stand Up. New York: W. W. Norton, 1980. While nonmathematical and written for the general audience, this book gives a technical introduction to the strength of materials and the principles of good structural design, including equilibrium.