# Lecture# 12: Rigid Bodies Rigid Body: For system Lecture# 12: Rigid Bodies Rigid Body: For system where forces are not concurrent F1 F3 F2 Coplanar Force System, Parallel Force System Non-concurrent etc. 02/26/20 1 Equilibrium: Section 4.1 Necessary Conditions are: R F 0

M o 0 moment of force about point o. Important the Point where forced is apply may rotate the body. P P 02/26/20 2 Moments: Tendency of Force to rotate the body. F d F d

o o Moment is a Vector: Magnitude + direction M o F d d moment arm, o moment center, d F direction 02/26/20 3 Can all Forces Produce Moment? Which of Forces produce moment about point o.

y F3 F2 x o F1 d1 d2 02/26/20 4 Section 4.2: Varignons 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 02/26/20 5 Summary: Moment is a vector: mag. + dir. Forces that passes through point of interest Mo=0. Direction indicated by: clocwise (-) counter clockwise (+)

02/26/20 6 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. Mo r F sin Mo Fo e o ro Direction: e

right-hand rule: 02/26/20 7 Sec. 4.3: Vector Representation of a Moment z Position Vector r rx i ry j rz k F Fx i Fy j Fz k o Fzk

r Fyj Fxi y Vector Force x Moment of a Force about Point o. M o r F 02/26/20 8 Vector Product in 3-D Space Moment about a point o. M o r F

z Mo z x o F y r y x

Cartesian Form M o M x i M y j M z k M o e 02/26/20 9 Components of Mo: The three components i j k M o r F rx Fx ry

Fy rz Fz Determinant of above matrix: M o rx Fz rz Fx i rz Fx rx Fz j rx Fy ry Fx k Components: M x rx Fz rz Fx 02/26/20 M y rz Fx rx Fz M z rx Fy ry Fx 10 Magnitude & Direction of Mo: Magnitude:

2 x 2 y Mo M M M 2 z Direction given by unit Vector e: e cos x i cos y j cos z k where Mx cos x Mo 02/26/20

cos y My Mo Mz cos z Mo 11 Example #1: Do problem in a 2-D space using above method. 02/26/20 12

Activity #1: Determine the moment of the force 760 lb about point A. y 760 12in 40o ANS:=11,871 in.lb ccw 02/26/20 x 10in 13 Example #2: 3D Space Find: (a) Moment about Mo

(b) Magnitude of Mo. Direction of Mo. 02/26/20 14 Activity#2 & #3 For following problem find (a) Mo=? (b) Magnitude Mo=? (c ) Direction =? 02/26/20 15 MAple Matrix A:=[x,y,z]

Mag:= evalf(sqrt(dotprod(A,A))) Matrix B:=[x,y,z] AXB:= crossprod(A,B); 02/26/20 16