What is the moment of a couple?

Short Answer:

The moment of a couple is the measure of the rotational or turning effect produced by a couple on a body. It is equal to the product of one of the forces of the couple and the perpendicular distance between their lines of action. Mathematically, it is expressed as M = F × d, where F is the magnitude of one of the forces and d is the perpendicular distance between them. The moment of a couple causes pure rotation without translation and is measured in Newton-meters (N·m).

Detailed Explanation :

Moment of a Couple

In Engineering Mechanics, the moment of a couple represents the rotational effect created by two equal and opposite forces acting on a body but not along the same line of action. Unlike a single force that may cause both translation and rotation, a couple only causes rotation. The forces in a couple act parallel to each other and are equal in magnitude but opposite in direction.

Because the two forces are equal and opposite, their linear (translational) effects cancel out, but they produce a turning effect or moment about a point or axis. This moment depends on two key factors:

  1. The magnitude of one of the forces, and
  2. The perpendicular distance between the lines of action of the two forces, known as the arm of the couple.

Hence, the moment of a couple can be defined as:
“The product of the magnitude of one of the forces and the perpendicular distance between their lines of action.”

Mathematically,

Where,

  • M = Moment of the couple (in N·m),
  • F = Magnitude of one of the forces (in N),
  • d = Perpendicular distance between the forces (in m).

Direction of Moment of a Couple

The moment of a couple has a direction that depends on whether the forces tend to rotate the body clockwise or anticlockwise.

  • If the couple tends to rotate the body in a clockwise direction, the moment is said to be a clockwise moment.
  • If the couple tends to rotate the body in an anticlockwise direction, the moment is said to be an anticlockwise moment.

The direction of the moment can also be determined using the right-hand rule:
If the fingers of the right hand curl in the direction of rotation caused by the couple, then the thumb points in the direction of the moment vector.

Derivation of the Moment of a Couple

Consider a rigid body acted upon by two equal and opposite parallel forces, F and –F, whose lines of action are separated by a perpendicular distance d.

Let O be any arbitrary point on the body. The moment of force F about point O is given by F × distance from O, and the moment of force –F about the same point is –F × (distance from O).

If the perpendicular distance between the forces is d, the total moment of the couple about point O becomes:

This shows that the moment of a couple is independent of the choice of the point O. No matter which point is chosen, the total moment remains the same. This is a key property of a couple and shows that its effect is purely rotational.

Units of Moment of a Couple

In the SI system, the moment of a couple is measured in Newton-meter (N·m).
In the CGS system, it is measured in dyne-centimeter (dyne·cm).
1 N·m = 10⁷ dyne·cm.

Properties of Moment of a Couple

  1. Produces Rotation Only:
    The couple causes only rotation and not translation because the two forces are equal and opposite.
  2. Independent of Reference Point:
    The moment of a couple is the same about any point in its plane. It does not depend on where the moments are measured.
  3. Magnitude:
    The magnitude of the moment is the product of one of the forces and the perpendicular distance between their lines of action.
  4. Direction:
    The direction of the moment of a couple depends on the tendency of rotation (clockwise or anticlockwise).
  5. Equivalent Couples:
    Two couples are said to be equivalent if they produce the same moment and have the same rotational effect on a body.
  6. Resultant of Two or More Couples:
    If multiple couples act on a body, their resultant moment is equal to the algebraic sum of the moments of all the couples.

Examples of Moment of a Couple

  1. Turning a Steering Wheel:
    When both hands apply equal and opposite forces on opposite sides of a steering wheel, they form a couple, and the wheel rotates about its center.
  2. Using a Wrench or Spanner:
    While tightening or loosening a nut, the force applied at the handle and the reaction at the nut create a couple, producing a turning effect.
  3. Opening a Bottle Cap:
    When you twist a bottle cap, equal and opposite forces applied by the hand form a couple that rotates the cap.
  4. Propeller or Fan Blades:
    The engine applies a couple to the fan blades, causing continuous rotation about the central axis.
  5. Bicycle Pedals:
    When you push one pedal down and the other up, a couple is formed that turns the crank and rotates the wheel.

Applications of Moment of a Couple

The concept of the moment of a couple is widely used in mechanical and structural engineering. Some applications include:

  • In mechanical systems, to calculate torque in engines, shafts, and gear mechanisms.
  • In structural engineering, to determine bending and twisting moments in beams.
  • In machine design, for analyzing levers, linkages, and rotating components.
  • In control systems, such as aircraft and ships, to produce desired rotations.

Importance in Engineering Mechanics

Understanding the moment of a couple is essential because it helps engineers analyze and design mechanical systems that involve rotational motion. It explains how torque is generated and how rotational equilibrium can be achieved in machines and structures. The concept is the foundation for studying torsion, bending, and rotational dynamics in mechanical design.

Example Problem

If two equal and opposite forces of 80 N act on a body with a perpendicular distance of 0.25 m between them, then the moment of the couple is:

Hence, the body experiences a moment of 20 N·m, which causes it to rotate about its axis.

Conclusion

In conclusion, the moment of a couple is the measure of the rotational effect produced by two equal and opposite forces whose lines of action are separated by a perpendicular distance. It is given by the product of one of the forces and the perpendicular distance between them. The moment of a couple is independent of the reference point and produces pure rotation without any translation. This principle is widely used in Engineering Mechanics to understand torque, rotational motion, and the equilibrium of mechanical systems.