## Examples For Practice

1. Determine the resultant of the 50-, 70-, SO- and 120-pound forces of Fig. 5.

 Ans. 260 pounds acting upwards 1.8 and 0.1 feet to the right of A and D respectively.

Fig. 32.

Fig. 33.

2. Determine the resultant of the 40-, 10-, 30- and 20-pound forces of Fig. 32.

 Ans. 80 pounds acting down l 5/8 feet from left end.

33. Algebraic Composition. The algebraic method of composition is best adapted to parallel forces and is herein explained only for that case.

If the plus sign is given to the forces acting in one direction, and the minus sign to those acting in the opposite direction, the magnitude and sense of the resultant is given by the algebraic sum of the forces; the magnitude of the resultant equals the value of the algebraic sum; the direction of the resultant is given by the sign of the sum, thus the resultant acts in the direction which has been called plus or minus according as the sign of the sum is plus or minus.

Ryerson Physical Laboratory.

Kent Chemical Laboratory.

TWO OF THE SCIENTIFIC LABORATORIES OF THE UNIVERSITY OF CHICAGO, CHICAGO, ILL.

Henry Ives Cobb, Architect.

For Other Buildings of the University, See Page 170.

If, for example, we call up plus and down minus, the algebraic sum of the forces represented in Fig. 32 is

- 40 + 10 - 30 - 20 + 50 - 15 = - 45; hence the resultant equals 45 pounds and acts downward.

The line of action of the resultant is found by means of the principle of moments which is (as explained in "Strength of Materials") that the moment of the resultant of any number of forces about any origin equals the algebraic sum, of the moments of the forces. It follows from the principle that the arm of the resultant with respect to any origin equals the quotient of the algebraic sum of the moments of the forces divided by the resultant; also the line of action of the resultant is on such a side of the origin that the sign of the moment of the resultant is the same as that of the algebraic sum of the moments of the given forces.

For example, choosing (.) as origin of moments in Fig. 32, the moments of the forces taking them in their order from left to right are

- 40 X 5 = - 200, + 10 X 4 = + 40, - 30 X 3 = - 90,

- 20 x 1 = - 20, - 50 x 2 = - 100, + 15 x 3 = + 45.*

Hence the algebraic sum equals

-200 + 40 - 90 - 20 - 100 + 45 = - 325 foot-pounds.

The sign of the sum being negative, the moment of the resultant about O must also be negative, and since the resultant acts down, its line of action must be on the left side of O. Its actual distance from O equals

325/45= 7.22 feet.

## Examples For Practice

1. Make a sketch representing five parallel forces, 200, 150 100, 225, and 75 pounds, all acting in the same direction and 2 feet apart. Determine their resultant.

* The student is reminded that when a force tends to turn the body on which it acts in the clockwise direction, about the selected origin, its moment is a given a plus sign, and when counter-clockwise, a minus sign.

 Ans. Resultant = 750 pounds, and acts in the same direction as the given forces and 4.47 feet to the . left of the 75-pound force.

2. Solve the preceding example, supposing that the first three forces act in one direction and the last two in the opposite direction.

 Ans. ' Resultant = 150 pounds, and acts in the same direction with the first three forces and 16.3 feet to the left of the 75-pound force.

Two parallel forces acting in the same direction can be compounded by the methods explained in the foregoing, but it is sometimes convenient to remember that the resultant equals the sum of the forces, acts in the same direction as that of the two forces and between them so that the line of action of the resultant divides the distance between the forces inversely as their magnitudes. For example, let F1 and F2 (Fig. 33) be two parallel forces. Then if R denotes the resultant and a and b its distances to F1 and F2 as shown in the figure,

R = F1 + F2, and a : b :: F2 : F1.

34. Couples. Two parallel forces which are equal and act in opposite directions are called a couple. The forces of a couple cannot be compounded, that is, no single force can produce the same effect as a couple. The perpendicular distance between the lines of action of the two forces is called the arm, and the product of one of the forces and the arm is called the moment of the couple.

A plus or minus sign is given to the moment of a couple according as the couple turns or tends to turn the body on which it acts in the clockwise or counter-clockwise direction.