MOMENT OF A FORCE – VECTOR FORMULATION • Moment of force F about point O can be expressed using cross product MO = r X F Magnitude • For magnitude of cross product, MO = rF sinθ • Treat r as a sliding vector. Since d = r sinθ, MO = rF sinθ = F (rsinθ) = Fd axis MOMENT OF A FORCE – VECTOR FORMULATION (cont) Direction • Direction and sense of MO are determined by righthand rule *Note: - “curl” of the fingers indicates the sense of rotation - Maintain proper order of r and F since cross product is not commutative MOMENT OF A FORCE – VECTOR FORMULATION (cont) Principle of Transmissibility • For force F applied at any point A, moment created about O is MO = rA x F • F has the properties of a sliding vector, thus M O = r1 X F = r2 X F = r3 X F MOMENT OF A FORCE – VECTOR FORMULATION (cont) Cartesian Vector Formulation • For force expressed in Cartesian form, M O r F rx j k ry rz Fx Fy Fz i • With the determinant expanded, MO = (ryFz – rzFy)i – (rxFz - rzFx)j + (rxFy –ryFx)k 112 3.5 Moment of a Force about a Specified Axis • For moment of a force about a point, the moment and its axis is always perpendicular to the plane • A scalar or vector analysis is used to find the component of the moment along a specified axis that passes through the point 113 114 Scalar Analysis • According to the right-hand rule, My is directed along the positive y axis • For any axis, the moment is M a Fd a • Force will not contribute a moment if force line of action is parallel or passes through the axis 115 MOMENT OF A FORCE ABOUT A SPECIFIED AXIS (cont) Vector Analysis • For magnitude of MA, MA = MOcosθ = MO·ua where ua = unit vector • In determinant form, uax uay uaz M a ua (r F ) rx ry rz Fx Fy Fz i M a (uax i ua y j uaz k ) rx Fx j ry Fy k rz Fz Ma M aua ua (r F)ua M a ua (r F) ua (r F) 117 118 119 120 121 rC 0.6i 0.3k M AB 0.894 0.447 u B rC F 0.6 0 600 200 0 0.3 53.64 300 122 Homework: 3-63 123 3.6 Moment of a Couple • Couple – two parallel forces – same magnitude but opposite direction – separated by perpendicular distance d • Resultant force = 0 • Tendency to rotate in specified direction • Couple moment = sum of moments of both couple forces about any arbitrary point 124 MOMENT OF A COUPLE (cont) Scalar Formulation • Magnitude of couple moment M = Fd • Direction and sense are determined by right hand rule • M acts perpendicular to plane containing the forces MOMENT OF A COUPLE (cont) Vector Formulation • For couple moment, M=rXF • If moments are taken about point A, moment of –F is zero about this point • r is crossed with the force F to which it is directed M = rB X F + rA X –F = (rB – rA) X F rB = rA + r A couple moment is a free vector, i.e., it can act at any point since M depends only on the position vector r directed between the forces not the position vectors rA and rB. 127 128 M R (r F) 129 MOMENT OF A COUPLE (cont) Resultant Couple Moment • Couple moments are free vectors and may be applied to any point P and added vectorially • For resultant moment of For more than 2 moments, two couples at point P, MR = ∑(r X F) MR = M1 + M2 Equivalent Couples • 2 couples are equivalent if they produce the same moment • Forces of equal couples lie on the same plane or plane parallel to one another 131 132 133 134 135 136 Homework: 3-70, 3-90 137 Equivalent System Point O is on the line of action of the force: Point O is not on the line of action of the force: 138 139 SIMPLIFICATION OF A FORCE AND COUPLE SYSTEM • An equivalent system is when the external effects are the same as those caused by the original force and couple moment system • External effects of a system is the translating and rotating motion of the body • Or refers to the reactive forces at the supports if the body is held fixed