Equilibrium of forces

When several forces act on a body, the body is said to be in equilibrium if it remains at rest or moves with constant velocity. For equilibrium, both the resultant force and the resultant moment (torque) on the body must be zero.

Principle of Moments

The principle of moments states that for a body in equilibrium, the sum of the clockwise moments about any point is equal to the sum of the anticlockwise moments about that same point.

Mathematically, for equilibrium:

$$\sum (\text{clockwise moments}) = \sum (\text{anticlockwise moments})$$

Conditions for Equilibrium

A body is in equilibrium if:

• The resultant (net) force on the body is zero.
• The resultant (net) moment (torque) about any point is zero.

This means the body will not accelerate linearly or rotate.

Vector Triangle for Coplanar Forces

If three coplanar forces act on a body in equilibrium, they can be represented in both magnitude and direction by the sides of a closed triangle drawn to scale.

To construct the vector triangle:

1. Draw the first force as an arrow to scale.
2. From the head of the first arrow, draw the second force to scale.
3. From the head of the second arrow, draw the third force to scale.
4. The triangle should close (the third arrow ends at the tail of the first), confirming equilibrium.

Example: Applying the Principle of Moments

Suppose a uniform beam is balanced horizontally on a pivot. A 20 N weight hangs 2 m from the pivot on one side, and a 10 N weight hangs 4 m from the pivot on the other side.

Calculate the moments about the pivot:

• Clockwise moment: $20\,\text{N} \times 2\,\text{m} = 40\,\text{Nm}$
• Anticlockwise moment: $10\,\text{N} \times 4\,\text{m} = 40\,\text{Nm}$

Since the moments are equal, the beam is in equilibrium.

Summary

• For equilibrium: net force = 0, net moment = 0.
• The principle of moments applies to rotational equilibrium.
• Three coplanar forces in equilibrium can be represented by a closed vector triangle.