Chapter 3 Rigid Bodies : Equivalent Systems of Forces Part -2

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## Chapter 3 Rigid Bodies : Equivalent Systems of Forces Part -2

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**Two forces F and -F having the same magnitude, parallel**lines of action, and opposite sense are said to form a couple. Couple**B**r rB A rA Moment of A Couple • Moment of the force F, - F MO= rA x F F • Moment of the force -F, MO= rB x (-F) O • Moment of the couple, MO= rA x F + rB x (-F) MO= (rA - rB) x F MO= rx F**Moment of the couple,**• The moment vector of the couple is independent of the choice of the origin of the coordinate axes, i.e., it is a free vector that can be applied at any point with the same effect. Moment of A Couple Where d is the perpendicular distance between the lines of action of F and -F**Problem 3.69 on page 116**• A piece of plywood in which several holes are being drilled has been secured to a workbench by means of two nails. Knowing that the drill applies a 12 Nm couple of moment on the plywood, determine the magnitude of resulting forces applied to the nails if they are located to • At A and B • At A and C**Problem 3.69 on page 116**• Nails are at A and B • 12 Nm = (0.45m) * F • ---> F = 26.67 N F=26.67 N -F= 26.67 N**Two couples will have equal moments if**Equivalent Couples • the two couples lie in parallel planes, and • the two couples have the same sense or the tendency to cause rotation in the same direction.**M**1 m 120 N 120 N M 120 N 120 N 1 m Equivalent Couples M 20 N 20 N 6 m**Non- Equivalent Couples**M 1 m 120 N 20 N 120 N M 20 N 6 m**Consider two intersecting planes P1 and P2 with each**containing a couple • Resultants of the vectors also form a couple • By Varigon’s theorem • Sum of two couples is also a couple that is equal to the vector sum of the two couples Addition of Couples**Couples Can Be Represented by Vectors**• A couple can be represented by a vector with magnitude and direction equal to the moment of the couple. • Couple vectors obey the law of addition of vectors. • Couple vectors are free vectors, i.e., the point of application is not significant. • Couple vectors may be resolved into component vectors.**Sample Problem 3.6 on page 113**Determine the components of the single couple equivalent to the couples shown.**Sample Problem 3.6 on page 113**SOLUTION: compute the sum of the moments of the four forces about D.**Sample Problem 3.6 on page 113**• ALTERNATIVE SOLUTION: • Attach equal and opposite 20 lb forces in the +x direction at A, thereby producing 3 couples for which the moment components are easily computed.**Attach equal and opposite 20 lb forces in the +x direction**at A Sample Problem 3.6 on page 113 • The three couples may be represented by three couple vectors,**Resolution of a Force Into a Force at O and a Couple**• Force vector F can not be simply moved to O without modifying its action on the body. • Attaching equal and opposite force vectors at O produces no net effect on the body. • The three forces may be replaced by an equivalent force vector and couple vector, i.e, a force-couple system.**Resolution of a Force Into a Force at O and a Couple**F F MO r A A O O Any force F acting at a point A of a rigid body can be replaced by a force-couple system at an arbitrary point O. It consists of the force F applied at O and a couple of moment MO equal to the moment about point O of the force F in its original position. The force vector F and the couple vector MO are always perpendicular to each other.**Sample Problem 3.7 on page 114**Replace the couple and force shown by an equivalent single force applied to the lever**Sample Problem 3.7 on page 114**First calculate the moment created by the couple Secondly resolve the given force into a force at O and a couple We move the force F to O and add a couple of moment about O**Problem 3.80 on page 118**• A 135 N vertical force P is applied at A to the bracket shown. The bracket is held by two screws at B and C. • Replace P with an equivalent force-couple system at B • Find the two horizontal forces at B and C that are equivalent to the couple of moment obtained in part a**-F= 225 N**F= 225 N Problem 3.79 on page 118 Find the two horizontal forces at B and C that are equivalent to the couple of moment obtained in part a**Reduction of a System of Forces to One Force and One Couple**Any system of forces can be reduced to a force-couple system at a given point O. First, each of the forces of the system is replaced by an equivalent force-couple system at O. Then all of the forces are added to obtain a resultant force R, and all of couples are added to obtain a resultant couple vector MO. In general, the resultant force R and the couple vector MO will not be perpendicular to each other.**Reduction of a System of Forces to One Force and One Couple**First, each of the forces of the system is replaced by an equivalent force-couple system at O. Then all of the forces are added to obtain a resultant force R, and all of couples are added to obtain a resultant couple vector MO.**For the beam, reduce the system of forces shown to (a) an**equivalent force-couple system at A, (b) an equivalent force couple system at B, and (c) a single force or resultant. Note: Since the support reactions are not included, the given system will not maintain the beam in equilibrium. Sample Problem 3.8 on page • SOLUTION: • Compute the resultant force for the forces shown and the resultant couple for the moments of the forces about A. • Find an equivalent force-couple system at B based on the force-couple system at A. • Determine the point of application for the resultant force such that its moment about A is equal to the resultant couple at A.**SOLUTION:**• Compute the resultant force and the resultant couple at A. Sample Problem 3.8 on page**Find an equivalent force-couple system at B based on the**force-couple system at A. • The force is unchanged by the movement of the force-couple system from A to B. The couple at B is equal to the moment about B of the force-couple system found at A. Sample Problem 3.8 on page