Moment Tanslation Tanslation + Rotation This otation tenency is known as moment M of foce (toque) xis of otation may be any line which neithe intesects no paallel to the line of action of foce Magnitue of moment epens on magnitue of an the length
Mathematical efinition M α Moment about axis - is efine as M = (N.m) Moment is a vecto Diection, nomal to - plane (ight han ule) xis - is calle moment axis Moment is a sliing vecto Μ= + 2-D xis becomes point Use sign convention to expess iection (+ fo CCW, fo CW)
The coss pouct M The moment of about point = α M = Magnitue M = sin(α) = Diection: nomal to the plane, ight han ule xyz axis have to satisfy the ight han ule; Sequence of an is impotant; iˆ ˆj = kˆ
Vaignon s theoem + The moment of a foce about any point y 1 x 2 1 y 2 x = The moment of the components of the foce about the same point M o = M o = 1 + 2 = ( 1 + 2) = + 1 2 Useful with ectangula components M o = x 2 - y 1 Can use with moe than 2 components
Sample (1) Calculate the magnitue of the moment about the base point of the 600-N foce.
Sample (2) The foce exete by the plunge of cyline B on the oo is 40 N iecte along the line B, an this foce tens to keep the oo close. Compute the moment of this foce about the hinge. What foce c nomal to the plane of the oo must the oo stop at C exet on the oo so that the combine moment about of the two foces is zeo?
Couple (1) a Couple is a moment pouce by two equal, opposite, an noncollinea foces. + M B B M = (a+) a = The moment of a couple has the same value fo all moment centes M = + ( ) = ( ) B M = Couple may be epesente as a fee vecto Diection of couple is nomal to the plane of two foce B
Couple (2) M M M M CCW couple CW couple Since couple is a fee vecto, the followings ae equivalent couples M M M M 2 /2 2
oce-couple systems given foce can be eplace by an equal paallel foce an a couple. B to the system Couple B B B M= oce-couple system No changes in the net extenal effect
Sample (3) The igi stuctual membe is subjecte to a couple consisting of the two 100-N foces. Replace this couple by an equivalent couple consisting of the two foces P an P, each of which has a magnitue of 400 N. Detemine the pope angle θ.
Sample (4) Replace the hoizontal 400-N foce acting on the leve by an equivalent system consisting of a foce at an a couple.
Sample (5) Calculate the moment of the 1200-N foce about pin of the backet. Begin by eplacing the 1200-N foce by a foce-couple system at point C.
Sample (6) T T C 45 1.6 m 0.8 m 0.8 m Detemine the combine moment M about point ue to the two equal tensions T = 8 kn in the cable acting on the pulley. Is it necessay to know the pulley iamete? 1.6 m B
Resultants The Resultant is the simplest foce combination which can eplace the oiginal foces without alteing the extenal effect on the boy 1 3 y 2 y y 2 3 (1) (2) R R 1 R 1 R 2 2 + 3 = R1 1 + 1 = R 3 R y 1y 1 x 1 θ 2 x R x 3x R = 1 + 2 + 3 +... = R x = x R y = y, R R 2 = ( x ) + ( y θ = tan 1 ( R y / R x x ) ) 2
Metho to get a esultant 1 3 1 2 2 3 M o =Σ( i i ) R = 1) Pick a point (easy to fin moment ams) 1 1 1 2 3 2 2 3 3 3) foces an moments R M o =R =M o /R 2) Replace each foce with a foce at point + a couple 4) Replace foce-couple system with a single foce
the cases 1 1 1 + 2 = 3 2 2 3 3 = 0 R M = = 3 M = 0 R =
Sample (7) Detemine the esultant of the fou foces an one couple that act on the plate shown.
Sample (8) Detemine an locate the esultant R of the two foce an one couple acting on the I-beam.
Sample (9) The five vetical loas epesent the effect of the weights of the tuss an suppote oofing mateials. The 400-N loa epesents the effect of win pessue. Detemine the equivalent foce-couple system at. lso, compute the x-intecept of the line of action of the system esultant teate as a single foce R.
Sample (10) Replace the thee foces acting on the bent pipe by a single equivalent foce R. Specify the istance x fom point to the point on the x-axis though which the line of action of R passes.