Chapter 2Moments, Couples, Forces, Equivalent Systems2.1 Position VectorThe position vector of a point P relative to a point O is a vector rOP=−→OP having thefollowing characteristics:• magnitude the length of line OP;• orientation parallel to line OP;• sense OP (from point O to point P).The vector rOPis shown as an arrow connecting O to P, as depicted in Fig. 2.1(a).The position of a point P relative to P is a zero vector.Let ı, j, k be mutually perpendicular unit vectors (cartesian reference frame) withthe origin at O, as shown in Fig. 2.1(b). The axes of the cartesian reference frameare x, y, z. The unit vectors ı, j, k are parallel to x, y, z, and they have the sensesof the positive x, y, z axes. The coordinates of the origin O are x = y = z = 0, i.e.,O(0, 0, 0). The coordinates of a point P are x = xP, y = yP, z = zP, i.e., P(xP, yP, zP).The position vector of P relative to the origin O isrOP= rP=−→OP = xPı + yPj + zPk.The position vector of the point P relative to a point M, M 6= O of coordinates(xM, yM, zM) isrMP=−→MP = (xP−xM)ı + (yP−yM)j + (zP−zM)k.The distance d between P and M is given byd = |rP−rM| = |rMP| = |−→MP| =q(xP−xM)2+ (yP−yM)2+ (zP−zM)2.12 2 Moments, Couples, Forces, Equivalent SystemsOxıjkyyP (xP,yP,zP)(xM,yM,zM)MrOP= rPrMPOPrOP(a)(b)Fig. 2.1 Position vector2.2 Moment of a Bound Vector About a PointDefinition. The moment of a bound vector v about a point A is the vectorMvA= rAB×v, (2.1)where rABis the position vector of B relative to A, and B is any point of line ofaction, ∆ , of the vector v (Fig. 2.2).The vector MvA= 0 if and only the line of action of v passes through A or v = 0.The magnitude of MvAis|MvA| = MvA= |rAB||v| sinθ ,where θ is the angle between rABand v when they are placed tail to tail. The per-pendicular distance from A to the line of action of v is2.2 Moment of a Bound Vector About a Point 3ABvMvA= rAB× vrABdθθBΔFig. 2.2 Moment of a bound vector v about a point Ad = |rAB| sinθ ,and the magnitude of MvAis|MvA| = MvA= d |v|.The vector MvAis perpendicular to both rABand v: MvA⊥ rABand MvA⊥ v. Thevector MvAbeing perpendicular to rABand v is perpendicular to the plane containingrABand v.The moment given by Eq. (2.1) does not depend on the point B of the line ofaction of v, ∆ , where rABintersects ∆ . Instead of using the point B the point B0(Fig. 2.2) can be used. The position vector of B relative to A is rAB= rAB0+ rB0Bwhere the vector rB0Bis parallel to v, rB0B||v. Therefore,MvA= rAB×v = (rAB0+ rB0B) ×v = rAB0×v + rB0B×v = rAB0×v,because rB0B×v = 0.4 2 Moments, Couples, Forces, Equivalent SystemsMoment of a Bound Vector About a LineDefinition. The moment MvΩof a bound vector v about a line Ω is the Ω resolute(Ω component) of the moment v about any point on Ω Fig. 2.3.ArvnΩΔFig. 2.3 Moment of a bound vector v about a line ΩThe MvΩis the Ω resolute of MvAMvΩ= n·MvAn= n·(r ×v) n= [n,r,v]n,where n is a unit vector parallel to Ω , and r is the position vector of a point on theline of action of v relative to a point on Ω .The magnitude of MvΩis given by|MvΩ|= |[n,r,v]|.The moment of a vector about a line is a free vector.If a line Ω is parallel to the line of action ∆ of a vector v, then [n,r,v]n = 0 andMvΩ= 0.If a line Ω intersects the line of action ∆ of v, then r can be chosen in such a waythat r = 0 and MvΩ= 0.If a line Ω is perpendicular to the line of action ∆ of a vector v, and d is theshortest distance between these two lines, then|MvΩ|= d|v|.Moments of a System of Bound VectorsDefinition. The moment of a system {S} of bound vectors vi,{S} = {v1,v2,...,vn} = {vi}i=1,2,...,nabout a point A is2.2 Moment of a Bound Vector About a Point 5M{S}A=n∑i=1MviA.Definition. The moment of a system {S} of bound vectors vi,{S} = {v1,v2,...,vn} = {vi}i=1,2,...,nabout a line Ω isM{S}Ω=n∑i=1MviΩ.The moments M{S}Aand M{S}Pof a system {S}, {S} = {vi}i=1,2,...,n, of boundvectors, vi, about two points A and P, are related to each other as follows,M{S}A= M{S}P+ rAP×R, (2.2)where rAPis the position vector of P relative to A, and R is the resultant of {S}.BiAvi{S}rAPrABiPrPBiFig. 2.4 Moments of a system of bound vectors, viabout two points A and PProof. Let Bia point on the line of action of the vector vi, rABiand rPBitheposition vectors of Birelative to A and P, Fig. 2.4. Thus,M{S}A=n∑i=1MviA=n∑i=1rABi×vi=n∑i=1(rAP+ rPBi) ×vi=n∑i=1(rAP×vi+ rPBi×vi)6 2 Moments, Couples, Forces, Equivalent Systems=n∑i=1rAP×vi+n∑i=1rPBi×vi= rAP×n∑i=1vi+n∑i=1rPBi×vi= rAP×R +n∑i=1MviP= rAP×R + M{S}P.If the resultant R of a system {S} of bound vectors is not equal to zero, R 6=0, the points about which {S} has a minimum moment Mminlie on a line calledcentral axis, (C A), of {S}, which is parallel to R and passes through a point Pwhose position vector r relative to an arbitrarily selected reference point O is givenbyr =R ×M{S}OR2.The minimum moment Mminis given byMmin=R ·M{S}OR2R.2.3 CouplesDefinition. A couple is a system of bound vectors whose resultant is equal to zeroand whose moment about some point is not equal to zero.A system of vectors is not a vector, therefore couples are not vectors.A couple consisting of only two vectors is called a simple couple. The vectors of asimple couple have equal magnitudes, parallel lines of action, and opposite senses.Writers use the word “couple” to denote the simple couple.The moment of a couple about a point is called the torque of the couple, M or T.The moment of a couple about one point is equal to the moment of the couple aboutany other point, i.e., it is unnecessary to refer to a specific point. The moment of acouple is a free vector.The torques are vectors and the magnitude of a torque of a simple couple is givenby|M| = d|v|,where d is the distance between the lines of action of the two vectors comprisingthe couple, and v is one of these vectors.2.4 Equivalence of Systems 790◦ABrv−v90◦dFig. 2.5 Couple of the vectors v and −v, simple coupleProof. In Fig. 2.5, the torque M is the sum of the moments of v and −v aboutany point. The moments about point A areM = MvA+ M−vA= r ×v + 0.Hence,|M| = |r ×v| = |r||v|sin(r,v) = d|v|.The direction of the torque of a simple couple can be determined by inspection:M is perpendicular to the plane determined by the lines of action of the two vectorscomprising the couple, and the sense of M is the same as that of r ×v.The moment of a