# 2.7. VECTORIAL REPRESENTATION OF MOMENTS AND COUPLES

2.7.1. Vectorial Representation of Moments. The moment of a force is a vector which is the product of distance and force. Hence in case of moment* of a force the cross-product of distance and force would be taken. Consider the Fig 2.31.

Let F = Force vector (Fxi + Fyj + Fzk)

r = Distance (or position) vector with respect to O

= xi + yj +zk

M = Moment of force about point 0

* A quantity which is the product of two vectors and the quantity is also a vector, then cross product of the two vectors will be taken. But if the quantity is scalar, then dot product is taken.

Mz = Moment of F about z-axis.

Also Mx, My, and Mz are known as scalar components of moment.

Problem 2.16. A force F=2i + 4j3k is applied at a point P(1,1, – 2). Find the moment of the force F about the point (2,-1, 2).

Sol.Given:

Force F = 2i + 4j3k

The position vector r of the point P w.r.t. O.             ,

= Position vector of point P

– Position vector of point O.

= (i + j2k) (2i – j + 2k)

r = (1- 2)i + [1 + (1)]j + [- 2 – 2]k

= -i + 2j – 4k

The moment M is given by

=  [(2)(- 3) – (- 4)(4)] i + [(- 4)(2) – (- 1) (- 3)]j + [(- 1)(4) – (2)(2)]k
=(- 6 + 16) i + (- 8 – 3)j + (- 4 – 4) k = 10i -11j – 8k. Ans.

2.7.2. Vectorial Representation of Couples. The moment produced by two equal, opposite parallel forces is known as couple.

Fig. 2.33. shows two equal opposite and parallel forces acting at points A and B. Let rA and rB are the position vectors of A and B with respect to O. The vector which joints B to A is represented by r.

The moment oftwo forces about point O is given by
Mo = rA × F- rB × F
= (rA – rB)× F
= r × F ( rA – rB=r)
The above equation shows that moment vector is independent of moment centre O.
M = r × F
This moment is known as couple.
The effect of couple is to produce pure rotation about an axis normal of the plane of force which constitute couple.

2.7.3. Scaler Components of a Moment. The moment of a force about any point O, is given by,

M = r × F

Where    Mx = Moment of F about x-axis = (yFzzFy),

My = Moment of F about y-axis = (zFxxFz) and

Mz = Moment of F about z-axis = (xFyyFx).

Also Mx, My and Mz are known as scaler components of Moment M.