Machinery's Handbook, 31st Edition
Force Systems 163 Composition of a Single Force and Couple.— A single force and a couple in the same plane or in parallel planes may be replaced by another single force equal in magnitude and parallel to the first force, at a distance from it equal to the moment of the couple divided by the magnitude of the force. The new single force is located so that the moment of the resultant about the point of application of the original force is of the same sign as the mo- ment of the couple. Fig. 12 illustrates this relationship. Forces N and − N are a couple. The moment of this couple is N ( ac + bc ). The resultant of N , − N , and a single force P is a force O , equal in mag- nitude to P and acting on an axis through a point c such that the moment of P and − N are equal to the moment of N . That is: ( P − N ) 3 ac = N 3 bc . Thus, ( ) ac P N ac bc P Moment of Couple = + =
P
O
N
a
c
b
— N
Fig. 12. Single Force and Couple Composition Algebraic Composition and Resolution of Force Systems.— The graphical methods given beginning on page 160 are convenient for solving problems involving force sys- tems in which all of the forces lie in the same plane and only a few forces are involved. If many forces are involved, however, or the forces do not lie in the same plane, it is better to use algebraic methods to avoid complicated space diagrams. Systematic procedures for solving force problems by algebraic methods are outlined beginning on page 163 . In connection with the use of these procedures, several terms applicable to force sys- tems in general must be defined. As has been illustrated, single force that produces the same effect upon a body as two or more forces acting together is called their resultant . The separate forces which can be so combined are called the components . Finding the resultant of two or more forces is called the composition of forces, and finding two or more components of a given force, the resolution of forces. Forces are said to be concurrent when their lines of action can be extended to meet at a common point; forces that are parallel are, of course, noncon- current . Two forces having the same line of action are said to be collinear . Two forces equal in magnitude, parallel, and in opposite directions constitute a couple . Forces all in the same plane are said to be coplanar; if not in the same plane, they are called non- coplanar forces. The resultant of a system of forces is the simplest equivalent system that can be deter mined. It may be a single force, a couple, or a noncoplanar force and a couple. This last type of resultant, a noncoplanar force and a couple, may be replaced, if desired, by two skewed forces (forces that are nonconcurrent, nonparallel, and noncoplanar). When the resultant of a system of forces is zero, the system is in equilibrium, that is, the body on which the force system acts remains at rest or continues to move with uniform velocity.
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