208 



COMPOSITION AND RESOLUTION OF FORCE. 



gle force may be assigned, which, acting on that point, will produce the same 

 result as the united effects of the other two. 



Let P, fig. 1, be the point on which the two forces act, and let their direc- 

 tions be P A and P B. From the point P, upon the line P A, take a length 



Fig. 1. 



P a, consisting of as many inches as there are ounces in the force P A ; and, 

 in like manner, take P b, in the direction P B, consisting of as many inches 

 as there are ounces in the force P B. Through a draw a line parallel to P B, 

 and through b draw a line parallel to P A, and suppose these lines meet at c. 

 Then draw PC. A single force, acting in the direction P C, and consisting 

 of as many ounces as the line P c consists of inches, will produce upon the 

 point P the same effect as the two forces P A and P B produce acting to- 

 gether. 



The figure P a c b is called, in geometry, a parallelogram ; the lines P a, P b, 

 are called its sides, and the line P c is called its diagonal. Thus the method 

 of finding an equivalent for two forces, which we have just explained, is gen- 

 erally called " the parallelogram of forces," and is usually expressed thus : " If 

 two forces be represented in quantity and direction by the sides of a parallelo- 

 gram, an equivalent force will be represented in quantity and direction by its 

 diagonal." 



A single force, which is thus mechanically equivalent to two or more other 

 forces, is called their resultant, and relatively to it they are called its compo- 

 nents. In any mechanical investigation, when the result is used for the com- 

 ponents, which it always may be, the process is called "the composition of 

 force." It is, however, frequently expedient to substitute for a single force two 

 or more forces, to which it is mechanically equivalent, or of which it is the re- 

 sultant. This process is called " the resolution of force." 



To verify experimentally the theorem of the parallelogram of forces is not 

 difficult. Let two small wheels, M N, fig. 2, with grooves in their edges to 

 receive a thread, be attached to an upright board, or to a wall. Let a thread be 

 passed over them, having weights, A and B, hooked upon loops at its extrem- 

 ities. From any part, P, of the thread between the wheels let a weight, C, be 

 suspended ; it will draw the thread downward, so as to form an angle, M P N, 

 and the apparatus will settle itself at rest in some determinate position. In this 

 state it is evident that, since the weight C, acting in the direction P C, balan- 

 ces the weights A and B, acting in the directions P M and P N, these two 

 forces must be mechanically equivalent to a force equal to the weight C, and 

 acting directly upward from P. The weight C is therefore the quantity of the 

 resultant of the forces P M and P N ; and the direction of the resultant is that 

 of a line drawn directly upward from P. 



To ascertain how far this is consistent with the theorem of " the parallelo- 



