Equilibrium of Rigid Body - Definition, Conditions for Equilibrium

By Deepak Yadav|Updated : August 26th, 2022

Before we go into the specifics of the equilibrium of rigid body, let's define a rigid body. A substance with a distinct shape unaffected by external force is said to be rigid. A rigid body is a collection of particles that are all uniformly spaced from one another and whose spacing cannot be altered. It is not necessary for each rigid body particle to behave exactly like every other particle. Every particle has a unique behavior that depends on the type of motion. The equilibrium of rigid body becomes important in this situation. What we will study next is how this equilibrium of rigid body impacts the entire particle system.

When an external force is applied, the spacing between particles in a rigid body remains constant. Therefore, our primary goal in studying the equilibrium of rigid bodies is to define the behavior of these constituent particles under various force or torque situations. We must take both translational and rotational motion into account because we are focused on the equilibrium of the rigid body in motion. Let's delve into the concepts of equilibrium of the rigid body.

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What Do You Mean by Equilibrium of Rigid Body?

The linear and rotational momentum are constant over time in the mechanical equilibrium of rigid body. According to this, a subject experiencing an external force neither experiences linear acceleration nor rotational acceleration. As a result, we can state:

  • Since the linear momentum is unchanged despite the change in time, a rigid body exhibits translational equilibrium if the total force acting on it is zero.
  • Because the angular momentum does not change over time, a rigid body exhibits rotational equilibrium if the total torque acting on it is zero.

Equilibrium of Rigid Body Definition

When all external forces or torques are equal to zero, a state of equilibrium is said to have occurred. This point could be anywhere close to the mass axis. An external force alters the linear momentum of a rigid body in translational motion. At the same time, the rigid body's angular momentum can be altered by external torque in rotational motion.

Dynamic Equilibrium 

Dynamic equilibrium is the state in which a moving body maintains its motion at a constant speed.

Static Equilibrium

Static equilibrium is the state in which the body is at rest and remains at rest with no change in the velocity or momentum of the particles.

Mention the Conditions for Equilibrium of Rigid Body

General requirements for a rigid body's equilibrium: A rigid body exhibits translational or rotational equilibrium when partially balanced. Take a rigid stick as an example subjected to an external force. A force of equal magnitude is applied in the opposite direction to the two ends of the stick while it is in rotational equilibrium, causing translational motion. However, when force is applied to two ends of a stick in a perpendicular direction, it results in a rotational motion rather than a translation of the stick. The stick is, therefore, in translational equilibrium.

A pair of forces is referred to as a couple or torque when they have the same magnitude but are directed in opposing directions and have different lines of action. A couple can create rotational motion but not translational motion. The two conditions for rigid body equilibrium are the following:

  • The first condition of equilibrium is that no net external forces operate on the body.
  • The second condition is that no net external torques from external forces exist.

Both of these conditions have to be met at the same time for equilibrium to occur.

Rigid Body in Motion

For rigid bodies to achieve equilibrium, where there is no linear or angular momentum, more than one force must act in the opposing direction. The vector sum of forces or torques is zero, and the linear and angular momentum are constant when the body is in equilibrium. Understanding the motions that a rigid body can go through is essential for understanding the equilibrium of a rigid body. A rigid body can move in two ways:

Translation Motion

The translational motion of the rigid body is the motion in which each rigid body particle moves linearly. On the other hand, a pure translational motion is one in which all of the rigid body's particles move at the same speed at any given instant and along a linear plane.

Rotational Motion

A rigid body can only move about the fixed point when it is fixed along a line or point, and this movement is known as a rotational motion. The axis of rotation of a rigid body is the fixed line it revolves around. The rotation of the earth's axis or the movement of a fan is the most common example of this motion. A rigid body rotates in a circular motion with the axis at the center of the movement of all its constituent particles.

Write Equations of Equilibrium of the Rigid Body

The total sum of moments around the fulcrum is zero in rotational equilibrium.

At rotational equilibrium,

d1F1 = d2 F2 or F1/F2 = d2/d1

Here,

  • F1 = load.
  • F2 = effort required to lift the object,
  • d1 = load arm and
  • d2 = effort arm. 

Center of Gravity

A rigid body is balanced at its center of gravity, resulting in a mechanical equilibrium between two rigid bodies. For instance, if a scale is balanced on the tip of a finger, the tip of the finger serves as the scale's center of gravity and its point of balance.

When the finger's tip has a force equal to and opposite from the force of gravity, the scales are balanced at that location. Gravity's force, which keeps the scale in equilibrium on the tip of the finger, causes the total torque to be zero at the point of balance.

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FAQ About Equilibrium of Rigid Body

  • The sum of the force components in the x and y directions, as well as the moments about the z-axis, are three possible equilibrium equations for a rigid body in a two-dimensional situation. The total of all of these will be zero.

  • The first condition of equilibrium is that there must be no net external forces acting on the body, and the second condition is that there must be no net external torques from external forces. For equilibrium to exist, these two conditions must be satisfied concurrently.

  • In essence, rigid entities have non-deforming mass with shape and size while particles only have non-deforming mass. When the length, size, rotation, and torque of the item must be taken into account, rigid body analyses are necessary.

  • A solid body that neglects deformation is referred to as a rigid body. In other words, regardless of the forces acting on a rigid body, the distance between any two points remains constant over time. A metal rod is an example of a rigid body.

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