Conditions of Equilibrium: Conditions of Translational and Rotational Equilibrium

Equilibrium is a state in which an object or system does not accelerate. This means that the net force and net torque acting on the object or system are zero.

• By releasing some amount of energy, every particle or system in this universe tends to achieve equilibrium.
• The stability of the system is defined as its capacity to withstand changes in its physical condition of rest or motion.
• A system is said to be stable if it can retain its equilibrium position or initial state, even after a small disturbance is applied to it.

There are two conditions of equilibrium. These are

1. First condition of equilibrium: It is also known as the condition for translational equilibrium. It states that for a body to be in translational equilibrium, the net force acting on it must be zero.
2. Second condition of equilibrium: It is also known as the condition for rotational equilibrium. It states that for a body to be in rotational equilibrium, the net torque acting on it must be zero.

Key Terms: Static equilibrium, Force, Torque, Angular momentum, Linear momentum, Translational equilibrium, Rotational equilibrium, Acceleration

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Static Equilibrium

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The static equilibrium of a particle or any system is defined as the condition when it is completely at rest i.e. no translational or rotational motion is observed. 

• The term 'static' means that the body is completely at rest, whereas 'equilibrium' means that all of the forces acting on the system are balanced.
• The nature of the forces is said to be completely balanced only when the forces that are directed upward are balanced and their effect is canceled out by the forces that are directed downward are the forces. 
• Similarly, forces directed to the left must be balanced by forces directed to the right. 
• This does not imply that the forces are identical in magnitude, but that they are completely balanced and hence cancel each other out.

Static equilibrium

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Conditions of Equilibrium

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To be in a state of equilibrium, a system must not undergo any kind of linear or rotational acceleration. That simply means that the net acting force, torque, or both must total to zero.

The two conditions of equilibrium are as follows

First Condition of Equilibrium

The first condition of equilibrium is also known as the condition for translational equilibrium.

A rigid body is said to be in translational equilibrium if it remains at rest or moving with constant velocity.

Let F₁, F₂, F₃,....... be the external forces acting on a rigid body of mass m, then according to Newton’s second law of motion,

Net external force = Rate of change of linear momentum

⇒ F₁ + F₂ + F₃ + ……… + F_n = dp/dt

⇒ ∑ F_ext = \(\frac{d}{dt}\)(mv)

If velocity v is constant, then ∑ F_ext = 0

Thus a body is said to be in translational equilibrium if the net external force acting on the body is zero i.e.

∑ F_ext = 0

Also, ∑ F_ext = \(\frac{dp}{dt}\)

The body is in translational equilibrium if ∑ F_ext = 0

Therefore, \(\frac{dp}{dt}\) = 0

⇒ p = constant

Thus a body is said to be in translational equilibrium, if its total linear momentum remains constant i.e. does not change with time i.e.

p = constant

The image below shows a stationary person in static equilibrium. As the image shows, the forces acting on him sum up to zero.

The car in the image below is in dynamic equilibrium since it moves at a constant velocity. There are horizontal and vertical forces, but there is no net external force in any direction.

The first condition of equilibrium

Second Condition of Equilibrium

The first condition of equilibrium is also known as the condition for rotational equilibrium.

A rigid body is said to be in rotational equilibrium if the body does not rotate with or rotate with constant angular velocity.

Let τ₁, τ₂, τ₃, ……… be the external torque acting on a rigid body, then net external torque is given by

∑ τ_ext = τ₁ + τ₂ + τ₃ + ..........

The equation of motion of a rotating body is given by

∑ τ_ext = \(\frac{dL}{dt}\)

The body will be in rotational equilibrium if angular momentum (L) is constant

⇒ \(\frac{dL}{dt}\) = 0

Therefore, ∑τ_ext = 0

Thus a rigid body is said to be in rotational equilibrium if the net external torque acting on the body is zero i.e.

∑ τ_ext = 0

 Also, we have ∑ τ_ext = \(\frac{dL}{dt}\)

The body is said to be in rotational equilibrium if ∑ τ_ext = 0

Therefore \(\frac{dL}{dt}\) = 0

⇒ L = constant

A body is said to be in rotational equilibrium if its total angular momentum is constant i.e. does change with time i.e.

L = constant

Two children balancing on a seesaw satisfy both conditions for equilibrium. In the figure, the lighter child is seated farther away from the pivot, creating a torque equivalent to that of the larger child.

Second condition of equilibrium

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Examples of Static Equilibrium

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Equilibrium is essential in the construction of rigid structures ranging from the floor system of the house to massive suspension bridges. 

• This is because these standing constructions are expected to remain in a state of equilibrium (particularly static equilibrium) under all estimated loading conditions. 
• Engineers must carefully plan and build structures that will not collapse, slide, or tilt in the worst-case situation. 
• In brief, they must ensure that their structures remain in equilibrium: no translational or rotational movements should be observed.

Here are some additional real-life Equilibrium examples that can easily observe in your surroundings. They are

• A book at rest on a table.
• A merry-go-round spinning at a constant speed.
• A person standing still on the ground.
• The Earth orbiting the sun.
• A car driving at a constant speed on a straight road.
• A seesaw with two children of equal weight sitting on opposite ends.
• A bicycle wheel rolling down the street.

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Things to Remember

• The equilibrium of a body is the state in which the net force and net torque acting on the body are zero.
• There are two types of equilibrium: static equilibrium and dynamic equilibrium.
• There are two conditions of equilibrium which are known as the first condition of equilibrium and the second condition of equilibrium.
• The first law of equilibrium states that a body will be in translational equilibrium if the net force acting on it is zero.
• The second equilibrium states that a body will be in rotational equilibrium if the net torque acting on it is zero.

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