What is equilibrium of a rigid body?

Short Answer:

The equilibrium of a rigid body is a condition in which the sum of all forces and the sum of all moments (torques) acting on the body are equal to zero. In this state, the body remains either at rest or continues to move with a constant velocity without rotation or acceleration.

In simple words, a rigid body is said to be in equilibrium when there is no unbalanced force or moment acting on it. This means the body does not experience any change in its motion or position under the influence of applied forces.

Detailed Explanation :

Equilibrium of a Rigid Body

The equilibrium of a rigid body is one of the most important concepts in engineering mechanics. It refers to the condition in which all external forces and moments acting on a body balance each other in such a way that there is no resultant force or moment.

In this state, the body remains in a steady condition — either completely at rest (static equilibrium) or moving with uniform velocity (dynamic equilibrium). The rigid body neither translates (moves linearly) nor rotates under these conditions.

Equilibrium analysis is crucial for mechanical engineers, as it helps in determining the stability, safety, and design of mechanical systems, structures, and machines such as bridges, cranes, beams, and frames.

Definition

The equilibrium of a rigid body can be defined as:

“A rigid body is said to be in equilibrium when the resultant of all the external forces and the resultant of all the external moments acting on it are both equal to zero.”

Mathematically, the conditions for equilibrium are:

 

Here,

  •  are the algebraic sums of all forces along the x, y, and z directions respectively.
  •  are the algebraic sums of all moments about the x, y, and z axes.

These conditions must be satisfied simultaneously for a rigid body to be in complete equilibrium.

Types of Equilibrium

The equilibrium of a rigid body can be classified mainly into two types based on motion and three types based on stability.

  1. Based on Motion:
  1. Static Equilibrium:
    When a rigid body remains completely at rest under the action of external forces, it is said to be in static equilibrium.
    Example: A beam supported on two ends and carrying a load remains stationary.
  2. Dynamic Equilibrium:
    When a body moves with constant velocity (linear or rotational) without acceleration, it is in dynamic equilibrium.
    Example: A car moving at constant speed on a straight road.
  1. Based on Stability:
  1. Stable Equilibrium:
    If a body returns to its original position after being slightly disturbed, it is in stable equilibrium.
    Example: A ball resting at the bottom of a bowl.
  2. Unstable Equilibrium:
    If a body moves away from its original position when slightly disturbed, it is in unstable equilibrium.
    Example: A ball balanced on the tip of a cone.
  3. Neutral Equilibrium:
    If a body remains in a new position after being disturbed, it is in neutral equilibrium.
    Example: A ball on a flat surface.

Conditions for Equilibrium of a Rigid Body

For a rigid body to be in complete equilibrium, the following conditions must be satisfied:

  1. Force Equilibrium:
    The vector sum of all the external forces acting on the body must be zero.

This ensures that there is no linear acceleration or translation of the body.

  1. Moment Equilibrium:
    The algebraic sum of the moments of all the external forces about any point must also be zero.

This ensures that there is no rotational acceleration or turning effect on the body.

When both these conditions are satisfied, the body remains in a completely balanced state.

Free-Body Diagram in Equilibrium Analysis

In engineering mechanics, equilibrium problems are solved by first drawing a free-body diagram (FBD) of the rigid body. This diagram represents:

  • All external forces acting on the body (including applied forces, reactions, and weights).
  • Points of application of forces.
  • The geometry of the system.

Using the FBD, the equations of equilibrium are written to solve for unknown forces and reactions.

For a two-dimensional system, the equations of equilibrium are:

For a three-dimensional system, there are six equations of equilibrium.

Examples of Equilibrium of a Rigid Body

  1. Beam Supported at Two Points:
    A horizontal beam with two supports and loads acting vertically is in equilibrium when the upward reactions at supports equal the total downward loads.
  2. Ladder Leaning Against a Wall:
    The ladder remains in equilibrium when the horizontal and vertical forces at the base and wall are balanced and the moments are zero.
  3. Bridge Structure:
    The forces in the truss members and supports are in equilibrium to carry the external loads safely.
  4. Crane Arm or Boom:
    The crane is designed so that all the forces acting on its structure are in equilibrium during lifting operations.

Importance of Equilibrium in Engineering

  1. Structural Design:
    Ensures that bridges, buildings, and machines remain stable under applied loads.
  2. Safety:
    Prevents failure or collapse of mechanical systems by maintaining balanced force and moment conditions.
  3. Machine Analysis:
    Helps in determining reactions, load distributions, and component stresses.
  4. Statics and Dynamics:
    Provides the foundation for solving both static and dynamic problems in mechanics.
  5. Efficiency and Performance:
    Proper equilibrium ensures smooth operation and longer life of mechanical systems.

Applications

  • Civil Engineering: Design of beams, trusses, arches, and columns.
  • Mechanical Engineering: Design of linkages, frames, shafts, and gear systems.
  • Automotive Engineering: Balance of forces in suspension systems.
  • Aerospace Engineering: Stability of aircraft and spacecraft under forces.
Conclusion

The equilibrium of a rigid body is a state in which all external forces and moments acting on the body are balanced, resulting in no net motion or rotation. The fundamental conditions for equilibrium are that the sum of all forces and the sum of all moments must be zero. Depending on motion, equilibrium may be static or dynamic, and depending on stability, it may be stable, unstable, or neutral. Understanding equilibrium is essential in mechanical engineering for designing safe, stable, and efficient mechanical and structural systems.