Le moment dû à un couple est défini par ???? = ± ???? ???? ???? s i n , où ???? est l'intensité de l'une des forces du couple, ???? est la longueur de la droite reliant les points à partir desquels les forces agissent et ???? est l'angle entre ? ???? et cette droite.
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2 10 2 10 lb ft M A 40
2 10 2 10 lb ft M A 40
2 10 2 10 lb ft M A 40
2 10 2 10 Effect of Couple applied at different points at the base of a Cantilever
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5. MOMENTS, COUPLES, FORCES SYSTEMS
& FORCE RESOLUTION (a) Translation (b) Translation & Rotation (c) RotationConcept of a Moment
When the Force is applied at the CG
When the Force is not applied at he CG
When the Force is not applied at the CG, & the body is hinged at the CG bodyCG of the body
Objective: To explain the concept of a Moment
If a Force P is applied at the midpoint of thefree, rigid, uniform object, it will slide the object such that every point moves an equal distance. The object is said to
translate. If the same force is applied at some other point as in second figure, then the object will both translate and rotate.If the point on the object is fixed against translation, (third figure) then the applied force causes the object to
rotate only.Objective: Explanation of the Concept of Moment -
c ontinued This tendency of a force to produce rotation about some point is called the Moment of a forceMoment of a Force
Objective: Definition of Moment in Statics
Moment of a Force
F dThe tendency of a force to produce rotation of a body about some reference axis or point is called the MOMENT OF A FORCE
M=Fxd Objective: An example to illustrate the definition of Moment in StaticsF= 25#
15"Lever arm
M= - F x d = -25 x 15= - 375 #-in
90 deg
d FMoment = Force x Perpendicular Distance = Fxd
Example One
: Closing the D oorExample Two:Tightening the NUT
Common Examples in the Application of the Concept of Moment Ob j ective: To ex p lain the conce p t of Moment in Statics with eve r y da y exam p lesSign Convention for Moments
Clockwise negative
Anti-clockwise positive
Objective: To illustrat
e the sign conventions for Moment in Statics F d M = - F d What is the moment at A for the Noodle Beam fixed at A and loaded by Force F at B? A BObjective: To illustrat
e that Moment is always Force x Distance, irrespective of the shape of the structureVarignon's
Theorem
y x d F Fx Fy FM=-F.d
M= -Fy.x Fx.y A AAccording to Varignon's
T heorem, a Force can be resolved into its components and multiplied by the perpendicular distances for easy calculation of the MomentObjective: To explain Varignon's
T heorem d F d F Fx Fy d cos d sin A A M ab out A= F x d sin cos d F d F d F x ySubstitute for Fx
and FyF x d =
sin sin cos cos d F d F d F 2 2 sin cos Fd Fd d F sin (cos 2 2 Fd Fd FdProof of Varignon's
Theorem
F d F d M abo u t A= - F x d d sin d cos FdFd d F d F d F d F M x y sin (cos sin sin cos cos sin cos 2 2 y F x F On the Left hand side the Moment is got directly by multiplying F times d.On the Right hand side it is proved the Moment is -F.d using Varignon's theorem.Proof Of Varignon's
Theorem
Ob j ective: To prove Vari g non's T heoremPlane of the couple
d F F F F Co ncept of a Co uple When you grasp the opposite side of the steering wheel and turn it, you are applying a couple to the wheel A couple is defined as two forces (coplanar) having the same magnitude parallel lines of action but opposite sense . Couples have pure rotational effects on the body with no capacity to translate the body in the vertical or horizontal direction. (Because the sum of their horizontal and vertical components are zero) d, arm of the couple Ob j ective: To ex p lain the conce p t of a Cou p le in Statics AA A 5' 10' 15' B C D 10lb 10lb 10lb 10lb 10lb 10lb 2' 2' 2' 2' 2' 2' lb ft M A 402 10 2 10 lb ft M A 40
2 10 2 10 lb ft M A 40
2 10 2 10 lb ft M A 40
2 10 2 10 Effect of Couple applied at different points at the base of a Cantilever