Equilibrium of a Particle in Physics with Key Concepts, Types, and Formula

The NCERT book’s section 4.8 mentions:

“Equilibrium of a particle in mechanics refers to the situation when the net external force on the particle is zero.”

What are the implications when a particle is in equilibrium?

• It satisfies Newton’s First Law of Motion. Being at equilibrium indicates there’s no net force that causes any acceleration on the object. 
• There are two states of equilibrium, one at rest (static) and the other while moving (dynamic). 

Learning about the equilibrium of a particle helps engineers determine whether objects, including bridges, buildings, hanging signs, and moving vehicles, will stay put (static) or keep moving (dynamic) at a steady speed. 

For a particle to be in mechanical equilibrium, the net external force from all directions acting on the particle must be zero

$\sum \vec{F}=0$

This is the first condition of equilibrium of a particle. That means, since there is no acceleration, i.e., a=0, then when suing F=ma, based on the special case of the 2nd Law of Motion, we still get to learn that there is no net external force. 

Now, since we are also considering the forces that occur from all sides, that is, coordinates, these forces from all sides must be balanced. 

If we consider the forces in each coordinate direction $x,y$ , both are balanced.

Mathematically, we can say that as

$\sum F_x=0,\mathrm{}\sum F_y=0$

In three dimensions, $\sum F_z=0$ will also apply. 

Equilibrium is of two types, or you could say, there are two states, which ideally follow the First Law of Motion. They also follow the mechanical equilibrium condition that the net external force is zero. 

• Static Equilibrium: This occurs for a particle at rest, where velocity is zero.  A classic example of static equilibrium is a book on a table that is at rest until you push it. 
• Dynamic Equilibrium: The particle moves with constant velocity, with no acceleration. Consider the basic example of a car cruising at a constant velocity on a straight road.

A free-body diagram (FBD) is essential for analysing equilibrium.

With it, you can show all common forces in mechanics acting on the particle in equilibrium. 

• Gravitational force ( mg ).
• Normal force ( N ).
• Tension (T).
• Frictional force ( f ).

For instance, have a look at this FBD. 

What this FBD is saying is that this is a generalised free-body diagram for a particle in equilibrium. To analyse the forces, we represent the particle as a dot. We draw each force as a vector arrow originating from the dot. This example shows a downward gravitational force (mg), an upward normal force (N), a tension force (T) to the right, and a frictional force (f) to the left.

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