Machinery's Handbook, 31st Edition
Force Systems 161 Parallelogram of Forces: If two forces applied at a point are represented in magnitude and direction by the adjacent sides of a parallelogram ( AB and AC in Fig. 4), their resul- tant will be represented in magnitude and direction by the diagonal AR drawn from the intersection of the two component forces.
R
R 1
P
B
B
R
R
C
A
C
A D
A
Q
Fig. 4. Fig. 6. If two forces P and Q do not have the same point of application, as in Fig. 5, but the lines indicating their directions intersect, the forces may be imagined as having been applied at the point of intersection between the lines (as at A ), and the resultant of the two forces may be found by constructing the parallelogram of forces. Line AR shows the direction and magnitude of the resultant, the point of application of which may be assumed to be at any point on line AR or its extension. Fig. 5. If the resultant of three or more forces having the same point of application is to be found, as in Fig. 6, first the resultant of any two of the forces ( AB and AC ) is found. Then, the resultant of the resultant just found ( AR 1 ) and the third force ( AD ) is found. If there are more than three forces, continue in this manner until the resultant of all remaining forces has been found. Parallel Forces: If two forces are parallel and act in the same direction, as in Fig. 7, then their resultant is parallel to both lines, is located between them, and is equal to the sum of the two components. The point of application of the resultant divides the line joining the points of application of the components inversely as the magnitude of the components. Thus, AB : CE = CD : AD The resultant of two parallel and unequal forces acting in opposite directions, Fig. 8, is parallel to both lines, is located outside of them on the side of the greater of the compo nents, has the same direction as the greater component, and is equal in magnitude to the difference between the two components. The point of application on the line AC produced is found from the proportion: AB : CD = CE : AE
B A
C E
E C
D
A D
F
F
B
Fig. 7. Fig. 8. Polygon of Forces: When several forces are applied at a point and act in a single plane, Fig. 9, their resultant may be found more simply than by the method just described, as fol lows: From the extreme end of the line representing the first force, draw a line representing the second force, parallel to it and of the same length and in the direction of the second
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