# Condition of Equilibrium in Mechanics

## Condition of Equilibrium:

Consider a body acted upon by a number of co-planar non-concurrent forces. A little consideration will show that as a result of these forces, the body may have any one of the following states:

1. The body may move in any one direction.

2. The body may rotate about itself without moving.

3. The body may move in any one direction and at the same time, it may also rotate about itself.

4. The body may be completely at rest.

Now, we have described the above-mentioned four states one by one.

1. If the body moves in any direction, it means that there is a resultant force acting on it. A little consideration will show that if the body is to be at rest or in equilibrium, the resultant forces causing movement must be zero. In other words, the horizontal component of all the forces (∑ H) and the vertical component of all the forces (∑ V) must be zero. Mathematically,

H = 0 and ∑ V = 0

2. If the body rotates about itself without moving. It means that a single-resultant couple is acting on it with no resultant force. A little consideration will show that if the body is to be at rest or in equilibrium, the moment of the couple causing rotation must be zero. In other words, the resultant moment of all the forces ∑ M) must be zero. Mathematically,

M = 0

3. If the body moves in any direction and at the same time it rotates about itself. It means that there is a resultant force and also a resultant couple acting on it. A little consideration will show that if the body is to be at rest or in equilibrium, the resultant force causing movements and the resultant moment of the couple causing rotation must be zero. In other words, the horizontal component of all the forces (∑ H), the vertical component of all the forces (∑ V) and the resultant moment of all the forces (∑ M) must be zero. Mathematically,

H = 0 ∑ V = 0 and ∑ M = 0

4. If the body is completely at rest, it necessarily means that there is neither a resultant force nor a couple acting on it. A little consideration will show that in this case the following conditions are already satisfied:

H = 0 ∑ V = 0 and ∑ M = 0

The above-mentioned three equations are known as the Condition of Equilibrium.