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PHYSICS

PART I. MECHANICS AND PROPERTIES
OF MATTER

Force, 1. Addition of forces, 2. The polygon of forces, 3. Forces in

equilibrium, 4. Examples of forces in equilibrium, 5. Vectors, 6. Com-

ponents of a vector, 7. The sailboat, 8. Forces that produce rotation, 9.

The moment of force, or torque, 10. Addition of moments, 11. Conditions

for complete equilibrium, 12. The center of gravity of a body, 13.

Couple, 14.

1. Force. One of the earliest things we learn in life is that it
requires a force to lift an object. We soon learn to make estimates
of the size or magnitude of the forces required; later, by means
of special instruments, such as a lever or a spring balance, we learn
to measure forces. Our earliest notions of force are associated
with that special type which we call weight. This, we notice,
always acts in a vertical direction. We soon learn, however, that
forces may act in any direction, and that if we wish to accomplish
something which requires a force we must pay as much attention
to the direction of the force as to its magnitude. In general, it
may be said that a force is completely described, or specified, only
when both its magnitude and direction are given.

In measuring the magnitude of forces various units are used.

In this book we shall generally use the pound and the dyne. The

first is commonly used in English-speaking countries; the second

is the international scientific unit. The dyne is an extremely small

unit:

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In this chapter will be given some of the methods and principles which must be known before a more detailed study of the relation of forces to other physical quantities is undertaken.

2. Addition of forces. When forces act in the same direction, they are added arithmetically. If to a weight of 10 pounds lying on a platform we add 15, and then 20 more, the total load will be 45 pounds. These forces can be added in this way only because they are in the same direction-they all act vertically downward. Forces which act in the same direction or in opposite directions are spoken of as parallel forces. In all cases parallel forces are added algebraically: those acting in one direction are called positive, and those in the opposite direction negative. Suppose that a given body has a number of forces acting on it-forces of 15, 25, and 30 pounds in one direction, and forces of 20 and 10 pounds in the opposite direction. These forces can be added as follows:

+ 15

+25

+30

20

10

+40

Hence the resultant force is one of 40 pounds acting in the direction called positive.

When the forces are not parallel, ordinary arithmetical or algebraic methods cannot be used. For example, suppose that one wished to find the sum, or resultant, of two forces making an angle of 60 degrees with each other-one of

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tude by the two lines A and B. The lengths of these lines are drawn to scale; that is, the line A is made twice as long as the line C, and B is made five times as long as C. The length of C represents 10 pounds. The length of the line C is entirely optional: any convenient length may be used. The parallelogram is completed by drawing the broken lines. The diagonal R, both in direction and magnitude, represents the sum, or resultant, of the two forces. The magnitude of R is determined by comparing its length with the length of the line C. In this way it is found that R represents a force of about 62 pounds. The diagonal R is sometimes called the geometrical sum of A and B.

In adding forces by the use of the parallelogram method two simple rules must be observed: (1) the two forces and the resultant are drawn from the same point, sometimes called the origin; (2) the arrows, which indicate the direction of the forces, should all point away from the starting-point, none toward it.

The parallelogram method may be used when there are more than two forces to be added. First, the resultant of any two of the forces is found by drawing a parallelogram. To this resultant the third force is added by the same method, a new resultant being found. This process is followed until all the forces have been. added. But when more than two forces are involved, the work is cumbersome. An improved method, called the polygon of forces, is explained in the following section.

3. The polygon of forces. It will be noticed by referring to Fig. 1 that it is not necessary to draw both the broken lines to determine the length and direction of R. All that is necessary is

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to draw two sides of the parallelogram. This is clearly seen in Fig. 2. Here B is drawn at the upper end of A, instead of at the lower end as in Fig. 1. The advantage of this method is more striking when one tries to find the resultant of several forces.

Let us try to add the four forces A = 6 pounds, B = 6 pounds, C = 4 pounds, and D = 5 pounds. These forces are drawn to scale and in the proper directions in Fig. 3. In Fig. 4 is shown the method of adding these four forces. The line A is drawn first. To the end of A is attached B, care being taken that B has the

proper length and direction. To the end of B is added C, and to that D. The line R, connecting the origin O and the end of D, is the resultant force. From the length, R is

found to be about 8.5 pounds.

It should be noted that in Figs. 2 and 4 the lines are so connected that two arrowheads do not point to any one junction, or corner, except O in the case of the resultant. When connected as shown in Fig. 5, incorrect results will be obtained. The student should satisfy himself of this by an actual trial of simple cases, such as the forces of Fig. 2.

A

B

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FIG. 4

B

A

FIG. 5

4. Forces in equilibrium. Whenever a number of forces acting at the same place have a resultant which is zero, the forces are said to be in equilibrium.. The lines which represent forces in equilibrium will form a closed figure when drawn by the method explained in the last section. For example, the forces represented in Fig. 6 are in equilibrium; their sum is zero. That they are in equilibrium may be proved as follows: The resultant of the forces AB and BC is a force AC acting from A toward C. If a third force equal and opposite to this resultant is added, the sum will be zero, and hence equilibrium will be obtained. But such a force has been included. It is the force CA, acting from C toward A. Hence the three forces AB, BC, and CA, which form the closed triangle ABC, are in equilibrium.

B

FIG. 6

C

In order that lines forming a closed A figure should represent forces in equilibrium, it is necessary that the forces act in the same sense around the figure. For example, the directions A to B, B to C, and C to A, in Fig. 6, are all in the same sense. In Fig. 4 the arrowheads do not all point in the same sense; but if the resultant R were replaced by a fifth force equal and opposite to R, the heads would all point in the same sense. Then the five forces would be in equilibrium, for they would meet the two conditions: (1) that the lines representing these forces form a closed figure, (2) that the arrowheads point in the same sense around the figure.

Whenever the resultant of a system of forces acting at a point is zero, lines representing these forces will form a closed polygon, with the direction of the forces in the same sense around the figure.

5. Examples of forces in equilibrium. A rope (Fig. 7) has a weight W in the middle. The problem can be stated thus: Given

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the inclination of the two parts of the rope, and the weight W of the load, to find the tension, or force, in the two parts of the rope. It is a matter of experience that when a body is at rest, the resultant of all the forces acting on it is zero. Hence the forces acting on A, shown in Fig. 8, are in equilibrium and, when properly placed together, will form a closed triangle (Fig. 9). In this problem the magnitude of W is known;

hence that line is drawn first, care being taken to get its direction and length cor

rect. The magnitudes of F1 and F2 are

1

2

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F

W

F

FIG. 9

ends of the line W in the correct directions until they intersect.

2

Then the lengths are carefully measured, and the magnitudes of the forces F1 and F2 determined from the scale of the figure. When the angle BAC of Fig. 7 is 120 degrees, the triangle of Fig. 9 is equilateral, and the forces F1, F2, and W are all equal. If the angle is nearly 180 degrees, a small load W will produce a large tension on the ropes AB and AC. This is the explanation of the great tension produced in taut wires when they become loaded with ice during an ice storm. The greater the slack in the wires,

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