MOMENT AND COUPLES

1. Introduction

In engineering mechanics, forces not only cause linear motion but can also produce rotation. The tendency of a force to rotate a body about a point or axis is known as the moment of a force. When two equal and opposite forces act at different points, they form a couple, which produces pure rotation without translation.
The concepts of moments and couples are fundamental in the analysis of levers, beams, machines, structures, and rotating systems.

2. Moment of a Force

Definition

The moment of a force about a point or axis is the product of the force and the perpendicular distance of its line of action from the point or axis.$$\text{Moment} = \text{Force} \times \text{Perpendicular Distance}$$Moment=Force×Perpendicular Distance $$M = F \times d$$M=F×d

Where:

• $F$F = applied force (N)
• $d$d = perpendicular distance (m)

Units of Moment

• SI unit: Newton–metre (N·m)

Types of Moments

1. Clockwise Moment – Tends to rotate the body in a clockwise direction
2. Anticlockwise Moment – Tends to rotate the body in an anticlockwise direction

By convention:

• Anticlockwise moment → Positive
• Clockwise moment → Negative (sometimes reversed as per sign convention)

3. Perpendicular Distance (Moment Arm)

• The moment arm is the shortest distance between the point of rotation and the line of action of the force.
• Larger the perpendicular distance, greater will be the turning effect.

4. Principle of Moments

Statement

For a body in equilibrium, the algebraic sum of the moments of all forces about any point is zero.

$$\sum M_{clockwise} = \sum M_{anticlockwise}$$

Applications

• Seesaws
• Levers
• Balancing of beams
• Structural equilibrium problems

5. Varignon’s Theorem (Theorem of Moments)

Statement

The moment of a resultant force about any point is equal to the algebraic sum of the moments of the individual forces about the same point.

This theorem simplifies moment calculations in complex force systems.

Moment of a Couple

$$M = F \times d$$M=F×d

Where:

• $F$F = magnitude of one force
• $d$d = perpendicular distance between the two forces (couple arm)

7. Properties of a Couple

1. A couple cannot be balanced by a single force
2. Couples with same magnitude and direction are equivalent
3. A couple can be moved anywhere on the body without changing its effect
4. Only another couple of equal magnitude and opposite direction can balance a couple

8. Difference Between Moment and Couple

Moment of Force	Couple
Caused by a single force	Caused by two equal and opposite forces
Depends on point of reference	Independent of reference point
Produces rotation and translation	Produces rotation only
Resultant force ≠ 0	Resultant force = 0

9. Torque

• Torque is the turning effect of a force about an axis.
• In mechanics, torque and moment are often used interchangeably.
• In rotating machinery, torque is commonly used.

$$\text{Torque} = F \times r$$Torque=F×r

Where $r$r is the radius from the axis of rotation.

10. Engineering Applications

• Design of levers, spanners, and wrenches
• Analysis of beams and frames
• Rotating machinery and engines
• Tightening and loosening of bolts
• Structural equilibrium calculations