# Moment of Couple & its Formula

Last updated on April 20th, 2023 at 02:21 am

**Moment of Couple** – this term in physics is based on an understanding of 2 concepts, the first one is ‘moment’ and the second one is ‘couple’. The **moment** in physics denotes the *turning effect*. A **couple in physics** is discussed in our post on *Couple – you can read it here*.

## Moment of Couple

Here, we will define the **moment of a couple**, and derive its formula as well.

### Moment of Couple – definition

**The moment of Couple is the cumulative turning effect produced by a ‘couple’ of forces and its magnitude is the product of either force of the couple and the perpendicular distance between the two forces of the couple (or couple arm length)**.

### derivation of the Moment of Couple formula

**Say, Two forces F and −F, having the same magnitude, parallel lines of action, and opposite directions, have formed a couple. **

Let us denote the position vectors of the points of application of F and – F by r_{A} and r_{B}, respectively. The moment M of the couple about O is the sum of the moments of F and of −F about O.

The sum of the moments of the two forces about O is:

r_{A}× F + r_{B}×(−F)=(r_{A}− r_{B})×F

Setting r_{A} − r_{B }= r, where r is the vector joining the points of application of the two forces, we conclude that the sum of the moments of F and −F about O is represented by the vector M = r × F

The vector M is called the **moment of the couple**. It is perpendicular to the plane containing the two forces, and its magnitude is M = r F sin θ = F d, where d is the perpendicular distance between the lines of action of F and −F, and θ is the angle between F (or −F) and r. The direction of M is defined by the right-hand rule.

**Moment of couple Formula**

The magnitude of Moment of Couple = Magnitude of either force of the couple x couple arm length