Equilibrium of Concurrent Forces

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Equilibrium of Concurrent Forces (ECF) is a term coined by Hervé Langevin. It appears in Langevin's 1911 paper "Applications des équations de la Dynamique Statistique'' where it is introduced as a new concept which generalizes the concept of force. The theory explains forces as the tendency for a system to change its path of least action so that its motion becomes continuous rather than a succession of jumps.

The concept was later developed further by Langevin, Ilya Prigogine and other physicists, culminating in the formulation of the concept of dissipative systems. In this formulation, equilibrium is achieved only when the total quantity of energy in the system becomes constant. The concept has now become ubiquitous in Physics.

It is possible to use Equilibrium of Concurrent Forces (ECF) to prove Lagrange's second Theorem. A simple way to state it is as follows: Two particles can approach a stationary point of action only if the force acting on the particles are equal and opposite. (See Langevin's 1911 paper) The reason is that forces will cause particles to move along the path of least action. If forces are not equal, two particles can approach a stationary point of action, but they will be pulled apart when they reach the point of stationary action.

History

Langevin's 1911 paper is the first to use the new term "equilibrium of concurrent forces" or "equilibrium of forces."  The paper was the first to describe motion as the path of least action. The concept can be traced back to the early work of Lagrange (1736) who used the phrase "path of least action". Lagrange also wrote of the path of least resistance, a phrase which would later be used by Ampere and Jacobi.

Generalizations

Equilibrium of concurrent forces is a much wider concept than that of equilibrium in Newton's mechanics, and the theory is sometimes called Lagrangian mechanics or (less commonly) the theory of Lagrange. 

However, the equilibrium of forces was originally meant to apply to constrained motion, and in this sense, it is not a generalization of the earlier theory. It is not a generalization of the more recent one of Hamiltonian mechanics.

In Lagrangian mechanics, two bodies can move through space in a constrained manner because there are certain constraints on their motion, which make it possible to define an action, or path of least action, and show that this is the path followed by the bodies. This method was invented by d'Alembert and Lagrange and is used in the first two chapters of Lagrange's Mechanics. 

In the case of Lagrangian mechanics, force constrains the motion of the bodies to give a well defined and calculable concept of action. For example, a string attached to a wall at each end is free to move but subject to the constraint that its motion is along the wall. Thus, its motion is constrained. 

The only free parameter is the total distance the string moves. The path of least action is that for which this distance is minimized. This is the path followed by the string and the wall is the "action" (or path) of the string on the wall.

In many classical mechanical systems, such as a rigid body, one force is absent, and there is only a single force on the system. For these systems, it is difficult to calculate the path of the body or the force acting on it, and this was a problem for Newton and Leibniz, who did not define or calculate the path of bodies or forces in their theory. 

A third person can observe the motion and infer that it is a path, but such inference is not in itself proof that the body or force is being acted upon by forces other than the motion, as required for the concept of action to be a measure of force. 

In Newton's analysis of the motion of a planet (and later in Lagrange's), the bodies and forces were not specified in a way that could be measured by others, except by observation, and the path of the body was deduced by the Newtonian theory as the path that is followed by a body in the absence of any forces other than gravity. Newton's theory does not apply to such systems where there are forces other than gravity acting on the system.

Newton's Law of Action and Reaction

In the classical system of classical mechanics, all forces act on the bodies in equal and opposite directions, so that the total work done on the system is zero. The Newtonian concept of action, therefore, provides a clear definition of force. Newton's second law of motion can be written, where m is the mass of the body and F is the force acting on it. For simplicity, Newton's action is assumed to be positive. In the formulation of classical mechanics, this law is applied to bodies whose motion is described by the positions r (vector) of the body at any time. An instantaneous change in position of the body is equivalent to the work done on the body by a force, and the magnitude of is where is the displacement of the body's mass point in the direction of. The sign of the work is arbitrary because in classical mechanics there is no preferred direction.

The work done by the force F applied on a body at position r. This is the Newtonian definition of action. In this formulation, the work done by the force (the change in the kinetic energy of the system) is a function of the instantaneous position of the body and the applied force; in this sense, the work is not a function of time.

The first term represents the increase in kinetic energy, and the second is the work done by the force F. The definition of action can be generalized to mechanical systems other than those of classical mechanics. In the Hamiltonian formulation, the system's state is given by the position and velocity of a body, and the total action is the work done on the system by a force.

What is the Equilibrium of Concurrent Forces?

To understand the Equilibrium of a concurrent force system, the first thing is to understand what equilibrium is. Equilibrium can be called the state of the body where all the forces acting on the body are balanced. This gets canceled out, making the net effect zero.

Concurrent forces refer to those forces which intersect through a common point. When it comes to concurrent forces, we can add them together as vectors and this gives the net resultant. A key point to remember is that acceleration is zero when a body is in a state of equilibrium. This is one of the conditions of Equilibrium for concurrent forces.

What are the Types of Equilibrium of Concurrent Forces?

There are two major types of equilibrium of concurrent forces. These include Static equilibrium and dynamic equilibrium. In static equilibrium, the state of the body is such that all the resultant forces add up to zero. The acceleration and velocity are both zero.

In dynamic equilibrium, the net result is zero, however, velocity is not. For example, a block resting on the floor can be said to be in static equilibrium. A block attached to a string in Simple Harmonic motion can be an example of dynamic equilibrium.

This is the simple way to define the Types of equilibrium of concurrent forces.

The Coplanar Forces in Equilibrium

Coplanar forces can be defined as forces that act on the same plane. There are certain conditions for Coplanar Forces to be in equilibrium. These include:

1. The sum of forces must be zero.

2. The sum of the moments of the forces about a point in the anticlockwise direction is equal to the sum of the moments of the forces about the same point in the clockwise direction.

It is important to note that there are various kinds of coplanar forces. In concurrent coplanar forces of equilibrium, the forces in one plane must intersect at one point. It can be calculated using graphical and algebraic methods.

The other types of coplanar forces are non-concurrent and parallel coplanar forces. In non-concurrent coplanar forces, the vectors do not meet at one point and are not parallel. In the case of parallel coplanar forces, the forces are all parallel to each other.

Hope you have received some basic information about the Equilibrium of concurrent forces.

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Solved Example

1. What are the conditions required for coplanar forces in equilibrium?

1. The Sum of all forces in X-direction should be equal to zero.

2. The Sum of all forces in the Y-direction should be equal to zero.

3. Both A and B.

4. Neither A nor B.

Ans: c) Both A and B.

Fun Fact:

The theory of equilibrium of concurrent forces can be explained using Newton's first law and second Law of motion.