Section A.20 Addition of Forces and Torques
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FIGURE A.3 The resolution of a force into its vertical and horizontal components.
A.20
Addition of Forces and Torques
Any number of forces and torques can be applied simultaneously to a givenobject. Because forces and torques are vectors, characterized by both a magnitude and a direction, their net effect on a body is obtained by vectorial addition.
When it is required to obtain the total force acting on a body, it is often convenient to break up each force into mutually perpendicular components. Thisis illustrated for the two-dimensional case in Fig. A.3. Here we have chosenthe horizontal x- and the vertical y-directions as the mutually perpendicularaxes. In a more general three-dimensional case, a third axis is required for theanalysis.
The two perpendicular components of the force F are Fx F cos θ
(A.23)
Fy F sin θ F is given by
F
F 2
x + F 2
y
(A.24)
When adding a number of forces (F1, F2, F3, . . .) the mutually perpendic ular components of the total force FT are obtained by adding the corresponding Appendix A Basic Concepts in Mechanics FT)x (F1)x + (F2)x + (F3)x + · · ·
(A.25)
(FT)y (F1)y + (F2)y + (F3)y + · · · The magnitude of the total force is FT FT)2x + (FT)2y
A.21
Static Equilibrium
A body is in static equilibrium if both its linear and angular acceleration arezero. To satisfy this condition, the sum of the forces F acting on the body,as well as the sum of the torques L produced by these forces must be zero;that is,
P
P F 0 and L 0
(A.27)
A.22
Work
In our everyday language, the word work denotes any types of effort whetherphysical or mental. In physics, a more rigorous definition is required. Here workis defined as the product of force and the distance through which the force acts.
Only the force parallel to the direction of motion does work on the object. Thisis illustrated in Fig. A.4. A force F applied at an angle θ pulls the object alongthe surface through a distance D. The work done by the force is Work F cos θ × D
(A.28)
A.23
Energy
