Lecture# 12: Rigid Bodies Rigid Body: For system where forces are not concurrent Coplanar Force System, Parallel Force System Non-concurrent etc. F1 F3 F2 7/23/2019

Equilibrium: Section 4.1 Necessary Conditions are: Important the Point where forced is apply may rotate the body. P P 7/23/2019

Moments: Tendency of Force to rotate the body. Moment is a Vector: Magnitude + direction d moment arm, o moment center, F d F d o o d F direction 7/23/2019

Can all Forces Produce Moment? Which of Forces produce moment about point o. y F3 F2 x o F1 d1 d2 7/23/2019

Section 4.2: Varignon’s Theorem The moment, M, of the Resultant force, R, is equal to the sum of the moments created by each single force. NOTE: ALL moments are taken at same point. Mo = RdR = F1d1 + F2d2 + F3d3 +….+Fndn Sign Convention for Moments: (+) counter clockwise (-) clockwise 7/23/2019

Summary: Moment is a vector: mag. + dir. Forces that passes through point of interest Mo=0. Direction indicated by: clocwise (-) counter clockwise (+) 7/23/2019

Vector Product of two vectors Vector Product of r and F must satisfy: Mo is perpendicular to plane of r & F. Magnitude of Mo given by. Direction: right-hand rule: o 7/23/2019

Sec. 4.3: Vector Representation of a Moment z Fzk Fyj r Fxi o y x 7/23/2019

Vector Product in 3-D Space Moment about a point o. z o y x 7/23/2019

Components of Mo: The three components Determinant of above matrix: 7/23/2019

Magnitude & Direction of Mo: Direction given by unit Vector e: where 7/23/2019

Example #1: Do problem in a 2-D space using above method. 7/23/2019

Activity #1: Determine the moment of the force 760 lb about point A. y ANS:=11,871 in.lb ccw y 760 12in 40o x 10in 7/23/2019

Example #2: 3D Space Find: (a) Moment about Mo (b) Magnitude of Mo. © Direction of Mo. 7/23/2019

Activity#2 & #3 For following problem find (a) Mo=? (b) Magnitude Mo=? (c ) Direction =? 7/23/2019

MAple Matrix A:=[x,y,z] Mag:= evalf(sqrt(dotprod(A,A))) Matrix B:=[x,y,z] AXB:= crossprod(A,B); 7/23/2019