What is meant by equilibrium?

A body is said to be in equilibrium if the system of forces acting on the body tends to keep the body at rest, in other words, there is no translation motion or net torque on the body. Under such conditions, the equilibrium conditions are to be applied to analyze the condition. The branch of mechanics that deals with this condition is known as 'statics'. Under equilibrium, there are no net forces on the body.

Different forces acting on a body

When more than one forces acts on a body, the body is said to be under a system of forces that may lie on a single plane or different planes. If the system of force is in a single plane, then the forces are termed co-planar forces. If they lie in different planes, they are termed non-co-planar forces or spatial forces.
When the line of action of a system intersects at a point, they are termed as concurrent forces; otherwise non-concurrent forces. A system of forces is collinear forces if it lies in the same line and parallel forces if they are parallel to each other.
A different system of forces will have a different effect on a rigid body that is the attribution of forces. For instance, if a body is acted upon by a system of concurrent external forces, the net effect on the body will tend to cause an acceleration in the body. If the body is acted upon by a system of non-concurrent external forces, there will be a net motion and a net torque acting on the body.

Resultant of a system of forces

When a system of forces acts on a body, it becomes necessary to study the effects of the forces on the body. Under such circumstances, the system of forces is replaced by a single force or an equivalent force, that produces the same effect on the body as produced by the system of forces. That single equivalent force is termed as the resultant of the forces.

When the resultant of the system of forces is zero, it means that the body is in a state of rest, or in a static condition, which is known by the term equilibrium. On the other hand, when the resultant turns out to be non-zero, they are not under equilibrium, and the body undergoes acceleration or under a state of uniform motion. During these conditions, the body is studied under the condition of dynamics.

Resultant of coplanar concurrent forces

Various methods are employed in determining the results of concurrent forces in a plane. Some of them are summarized below,

Cosine law: Cosine law can be stated as the mathematical statement of the parallelogram law. The cosine law relates the lengths of sides of a triangle to the cosine of its angles.

The cosine law is mathematically written as,

$R=\sqrt{F_1^2+F_2^2+2F_1{F}_2\cos \theta }$

Where,  $R$ = resultant of the system of two forces,  $F_1$ and  $F_2$

$\theta$ = angle between the two forces

Parallelogram law: The parallelogram law states that when two concurrent forces acting on a body are represented by the two adjacent sides of a parallelogram, the diagonal passing through the point of concurrency gives the resultant of the two forces.

Triangle law: The triangle law states that if two forces acting simultaneously on a body are represented by the sides of a triangle, then the third side can be represented as the resultant of the two forces.

Polygon law: When several concurrent forces acting simultaneously on a body can be represented by the sides of a polygon, the result is given by the closing side of the polygon.

Equilibrium conditions

The system of forces acting on a rigid body tends to cause a displacement in the body or tends to cause a net torque on the body. To analyze a body for equilibrium, first, a sketch of the body, irrespective of the supports, is to be made. In other words, a free-body diagram needs to be made. Then the active and reactive forces are to be included.

Active forces or external forces

These kinds of forces depend on the environment in which the body is placed. For instance, if the body is kept in an electric field, the body will experience a net external force of attraction and repulsion. Similarly, external forces are experienced by the body if the body is kept in gravitational and magnetic fields. If a body is kept in a conservative force field such that the forces are equal and opposite in nature, the body will experience a net torque or will undergo rotation.

Reactive forces

Reactive forces are generally induced if the motion of the body is constrained by another body or by any fixed supports. These reactive forces are developed on account of Newton's third law of motion. Hence, the reactive forces are always equal and opposite in nature.

There are many types of reactive forces, some of them are,

• Normal reaction: Whenever two bodies are in contact with smooth surfaces, the normal reaction forces are induced.
• Tensile forces: These kinds of forces are developed whenever any rigid body is tied to an extensible string (the string has minimum elasticity). For instance, a pendulum bob tied to a string, a rope tied to a pulley, the drive belt of an engine, and so on.
• Restoring forces: Whenever a body has a certain amount of elasticity, upon deformation, the body tries to resist that deformation by inducing a restoring force.
• Tension or compression members: In most structural engineering applications, the bodies of the members remains hinged at an angle, at a certain pivot point at the connections. Under such conditions, the forces act at the axes of such members. The members can thus be said to be either compression or tension.
• Support reactions: In some cases of rigid bodies, to constrain the rotational and translational motions, certain supports are provided. These supports in turn induce support reactions in the body.

There are three equilibrium conditions of a rigid body in a three-dimensional spatial space, they are,

• The algebraic sum of all the forces along the x-direction should be zero, $\sum _{}{F}_x=0$
• The algebraic sum of all the forces along the y-direction should be zero, $\sum _{}{F}_y=0$
• The algebraic sum of all the moments acting in the rigid body must be equal to zero,  $\sum _{}M=0$

The algebraic summation of all the forces along the x-direction equated to zero is the first condition that must be looked for while attempting a problem in statics. The rest of the equilibrium condition depends on the system of forces acting on the rigid body. 

Statically determinate and indeterminate structures

The three equilibrium conditions explained above are valid or can be applied easily if the number of reactions in the rigid body equals the number of equations governing the equilibrium conditions. Such structures are known as statically determinate structures. If the number of reactions exceeds the number of equations governing the equilibrium conditions, then the structures are known as statically indeterminate structures. Using the traditional method by applying the equilibrium conditions will not lead to a solution. In fact, an additional equation, known as the compatibility condition, needs to be applied to arrive at the solutions.

The nature of structures, that is, whether it is statically determinate or statically indeterminate depends on the type of loading conditions, a few of which are discussed below,

• Concentrated or point loads: The forces that act at a particular point in the rigid body. The weight of the body is then assumed to be concentrated at a particular point.
• Uniformly distributed loads (UDL): When the loads are evenly distributed over the entire length of the rigid body, such loads are termed as UDL. The loads are represented by force per unit length.
• Uniformly varying loads (UVL): The loads which are evenly distributed but vary linearly are termed UVL.

For both the UDL and UVL, the loads are replaced by a single equivalent load called the resultant force.

Context and Applications

The study of the equilibrium of forces on a rigid body is a preliminary task in analyzing structures in

• Bachelors in Technology (Civil Engineering, Mechanical Engineering)
• Masters in Technology (Civil Engineering, Mechanical Engineering)
• Bachelors in Science (Applied Science, Applied Mechanics, and Applied Mathematics)
• Masters in Science (Applied Science, Applied Mechanics, and Applied Mathematics)

Practice Problems

1. Which of the following branches of mechanics deals with the equilibrium of bodies?

1. Statics
2. Dynamics
3. Kinematics
4. Kinetics

Correct option- a

Explanation: Statics is a branch of mechanics that deals with the equilibrium conditions of the bodies at rest.

2. When a system of concurrent forces lies in a single plane, which of the following options holds true?

1. The forces are co-planar.
2. The forces are non-co-planar.
3. The forces are collinear.
4. None of these

Correct option- a

Explanation: When a system of concurrent forces lies at a single plane or common plane, the system of forces is termed as coplanar forces.

3. Which of the following forces are induced in a rigid body if it is connected by an inextensible string?

1. Tensile forces
2. Normal reaction
3. Support reaction
4. All of these

Correct option- a

Explanation: When a body is connected by an inextensible string, wire, rope, or belt, it always experiences a tensile pull or a tensile force.

4. How many equations are there to satisfy the conditions of equilibrium for a statically indeterminate structure?

1. Three
2. Four
3. Two
4. One

Correct option- b

Explanation: To analyze a statically indeterminate structure, a fourth equation called the compatibility equation, must be satisfied along with the three equilibrium conditions.

Q5) When a system of forces is represented by a single equivalent force, which of the following options is/are correct?

1. The equivalent force is known as the resultant.
2. The equivalent force is a resultant moment.
3. The equivalent force is a resultant of only two systems of forces.
4. None of these

Correct option- a and b

Explanation: When the system of forces is represented by a single equivalent force, the force is termed as resultant. If the system of forces is only pointing loads, then the results will only be a single point force. If the system consists of moments then the results must be an equivalent moment.

Related concepts

• Sine law
• Lami's theorem
• Application of equilibrium equation in three-dimensional (x, y, and z-axis) coordinate system.
• The inertia of a body