CE Statics Lecture 12

MOMENT OF A FORCE ABOUT A SPECIFIED AXIS Scalar Analysis Mo = (20 N) (0.5 m) = 10 N.m (F tends to turn about the Ob axis) How can we determine the component of Mo about the y-axis (My)? If  is known, then My = Mo cos (  ) It is clear that in order to find My Determine Mo Resolve Mo along the y-axis A F = 20 N 0.5 m O MOMO dy = 0.4 m b  dx = 0.3 m x y z MyMy

Generally, to find My directly, it is necessary to find the perpendicular distance from the line of action of the force to the y-axis (da), then: Ma = F (da) where: a is the aa-axis da is the perpendicular distance from F line of action to aa- axis Note: F will not have a moment about an axis if F is parallel to that axis or passes through it.

Vector Analysis The two steps performed with scalar analysis can be performed with vector analysis, therefore: Mo = rA  F = (0.3 i j)  (-20 k) = (-8 i + 6 j) N.m The component of Mo along the y- axis is determined from the dot product, then: My = Mo. u = (-8 i + 6 j). j = 6 N.m A F = (- 20 k) N r OA O MOMO 0.4 m b  0.3 m x y z MyMy u = j

Generally, if a body is subjected to force F acting at point (A), what is Ma?? A F r OA O M O = r  F b a b a Ma 

(1) Calculate Mo ( Mo = r OA  F ) The moment axis is perpendicular to the plane containing F and r (say bb) (2) Ma is the component along aa, where its magnitude is found from the dot product, ( Ma = Mo cos (  ) = Mo. u a ), where u a is the unit vector defining the direction of aa, then: Ma = (r  F). u a Since the dot product is commutative (A. B = B. A), then: Ma = u a. (r  F)

The combination of the dot and cross products is called the triple scalar product. i j k Ma = (u ax i + u ay j + u az k ). rx ry rz Fx Fy Fz Or simply,

u ax u ay u az Ma = u a. (r  F) = rx ry rz Fx Fy Fz If Ma is +ve, then it has the same direction as ua and if Ma is –ve, then it is in the opposite direction of u a. Whence Ma was calculated, Ma can be expressed in Cartesian vector form by: + Ma = Ma u a = [ u a. (r  F) ] u a The resultant moment of a system of forces about the axis is: Ma =  [ u a. (r  F) ] = u a.  (r  F)