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II. CONCURRENT FORCES; COMPOSITION AND RESOLUTION.

9. Graphical Composition of Two Concurrent Forces. If two forces are represented in magnitude and direction by AB and BC (Fig. 3), the magnitude and direction of their resultant is represented 6y AC. This is known as the "triangle law." The line of action of the resultant is parallel to AC and passes through the point of concurrence of the two given forces; thus the line of action of the resultant is ac.

The law can be proved experimentally by means of two spring balances, a drawing board, and a few cords arranged as shown in Fig. 4. The drawing board (not shown) is set up vertically, then from two nails in it the spring balances are hung, and these in turn support by means of two cords a small ring A from which a ' heavy body ( not shown) is suspended. The ring A is in equilibrium under the action of three forces, a downward force equal to the weight of the suspended body, and two forces exerted by the upper cords whose values or magnitudes can be read from the spring balances. The first force is the equilibrant of the other two. Knowing the weight of the suspended body and the readings of the balances, lay off AB equal to the pull of the right-hand upper string according to some convenient scale, and BC parallel to the Fig. L left-band upper string and equal to the force exerted by it. It will then be found that the line joining A and C is vertical, and equals (by scale) the weight of the suspended body. Hence AC, with arrowhead pointing down, represents the equilibrant of the two upward pulls on the ring; and with arrowhead pointing up, it represents the resultant of those two forces.

Notice especially how the arrowheads are related in the tri angle (Fig. 3), and be certain that you understand this law before proceeding far, as it is the basis of most of this subject.

Examples. Fig. 5 represents a board 3 feet square to which forces are applied as shown. It is required to compound or find the resultant of the 100- and 80 pound forces.

First we make a drawing of the board and mark upon it the lines of action of the two forces whose resultant is to be found, as in Fig. 6. Then by some conven ient scale, as 100 pounds to the inch, lay off from any convenient point A, a line AB in the direc tion of the 100-pound force, and make AB one inch long, repre senting 100 pounds by the scale. Then from B lay off a line BC

in the direction of the second force and make BC, 0.8 of an inch long, representing 80 pounds by the scale. Then the line AC, with the arrow pointing from A to C, represents the magnitude and direction of the resultant. Since AC equals 1.06 inch, the result ant equals 1.06 X 100 106 pounds.

The line of action of the resultant is ac, parallel to AC and ing through the intersection of the lines of action (the point of concurrence) of the given forces. To complete the notation, we mark these lines of action ab and Lc as in the figure.

1. Determine the resultant of the 100- and the 120-pound forces represented in Fig. 5.

/The magnitude is 194 pounds; the force Ans. acts upward through A and a point 1.62 feet to the right of D.

2. Determine the resultant of the 120- and the 160-pound forces represented in Fig. 5.

The magnitude is 200 pounds; the force Ans. acts upward through A and a point 9 inches below C.

io. Algebraic Composition of Two Concurrent Forces. If the angle between the lines of action of the two forces is not 90 degrees, the algebraic method is not simple, and the graphical is usually preferable. If the angle is 90 degrees, the algebraic meth od is usually the shorter, and this is the only case herein explained.

Let and F2 be two forces acting through some point of a body as represented in Fig. 7a. AB and BC represent the magni tudes and direction of and F2 respectively; then, according to the triangle law (Art. 9), AC represents the magnitude and direc tion of the resultant of and F,, and the line marked R (parallel to AC) is the line of action of that resultant. Since ABC is a right triangle, Now let R denote the resultant. Since AC, AB, and BC represent and F2 respectively, and angle CAB = o, = + F,'; or R = + F,2; and, tan x = By the help of these two equations we compute the magni tude of the resultant and inclination of its line of action to the force Example. It is required to determine the resultant of the 120- and the 1150-pound forces represented in Fig. 5.