# C_2 - Chapter 2 Moments Couples Forces Equivalent Systems...

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Chapter 2 Moments, Couples, Forces, Equivalent Systems 2.1 Moment of a Vector About a Point Deﬁnition . The moment of a bound vector v about a point A is the vector M v A = r AB × v , (2.1) where r AB is the position vector of B relative to A , and B is any point of line of action, Δ , of the vector v (Fig. 2.1). The vector M v A = 0 if and only the line of action of v passes through A or v = 0 . The magnitude of M v A is | M v A | = M v A = | r AB || v | sin θ = r AB v sin θ , A B v M v A = r AB × v r AB d θ θ B ± Δ Fig. 2.1 Moment of a vector v about a point A 1

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2 2 Moments, Couples, Forces, Equivalent Systems where θ is the angle between r AB and v when they are placed tail to tail. The per- pendicular distance from A to the line of action of v is d = | r AB | sin θ = r AB sin θ , and the magnitude of M v A is | M v A | = M v A = | v | d = vd . The vector M v A is perpendicular to both r AB and v : M v A r AB and M v A v . The vector M v A being perpendicular to r AB and v is perpendicular to the plane containing r AB and v . The moment given by Eq. (2.1) does not depend on the point B of the line of action of v , Δ , where r AB intersects Δ . Instead of using the point B the point B 0 (Fig. 2.1) can be used. The position vector of B relative to A is r AB = r AB 0 + r B 0 B where the vector r B 0 B is parallel to v , r B 0 B || v . Therefore, M v A = r AB × v = ( r AB 0 + r B 0 B ) × v = r AB 0 × v + r B 0 B × v = r AB 0 × v , (2.2) because r B 0 B × v = 0 . The moment of a vector about a point (which is also the moment about a deﬁned axis through the point) is a sliding vector whose direction is along the axis through the point. Next, using MATLAB R ± , it will be shown the validity of Eq. (2.2). Three points A , B , and C are deﬁned by three symbolic position vectors r A , r B , and r C : syms x_A y_A z_A x_B y_B z_B x_C y_C z_C real r_A = [x_A y_A z_A]; r_B = [x_B y_B z_B]; r_C = [x_C y_C z_C]; The vector v is v = r C - r B , or in MATLAB: v = r_C - r_B; The line of action of the vector v is deﬁned as the line segment BC . A generic point B 0 (in MATLAB Bp ) divides the line segment joining two given points B and C in a given ratio. The position vector of the point Bp is r Bp : syms k real % k is a given real number r_Bp = r_B + k * (r_C-r_B); The moment of the vector v with respect to A is calculated as r AB × v , r AB 0 × v , and r AC × v , or with MATLAB: Mv_AB = cross(r_B-r_A, v); % r_AB x v Mv_ABp = cross(r_Bp-r_A, v); % r_ABp x v Mv_AC = cross(r_C-r_A, v); % r_AC x v
2.1 Moment of a Vector About a Point 3 To prove that M v A = r AB × v = r AB 0 × v = r AC × v the following MATLAB com- mands are used: simplify(Mv_AB) == simplify(Mv_ABp) simplify(Mv_AB) == simplify(Mv_AC) To represent the vectors r AB , r AB 0 r , v and M v A the following numerical data are used: x A = y A = z A = 0 , x B = 1 , y B = 2 , z B = 0 , x C = 3 , y C = 3 , z C = 0 , and k = 0 . 75. The numerical values for the vectors r A , r B , r C , r Bp , v , Mv AB , Mv ABp , and Mv AC are calculated in MATLAB with: slist={x_A,y_A,z_A, x_B,y_B,z_B, x_C,y_C,z_C, k}; nlist={0,0,0, 1,2,0, 3,3,0, .75}; rA = double(subs(r_A,slist,nlist)) rB = double(subs(r_B,slist,nlist)) rC = double(subs(r_C,slist,nlist)) rBp = double(subs(r_Bp,slist,nlist)) V = double(subs(v,slist,nlist)) MvA = double(subs(Mv_AB,slist,nlist))

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C_2 - Chapter 2 Moments Couples Forces Equivalent Systems...

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