9. Representation of Forces. Forces have a certain magnitude, act in a certain direction, and have a definite point of application. If a man, for example, attaches a rope to a log and pulls on the rope, his pull may be measured in pounds; it acts along the rope, and it has a point of application which is the same as the point of attachment of the rope to the log. It has been found convenient, for the purpose of analysis, to represent forces by arrows (vectors of Art. 8), the length of the arrow representing the magnitude of the force and the direction of the arrow giving the direction in which it acts. Thus, a 10-pound force, acting in a direction 30° with the horizontal, is represented by an arrow drawn in the same direction and having its point of application in the body and having a length representing 10 lb. (In this case, if 2 lb. represents 1 in., the length of the arrow is 5 in.) The line along which a force acts will be referred to as its line of action.

10. Concurrent Forces. When two or more forces act upon the same point of a body, their lines of action are concurrent, and the forces are known as concurrent forces.

11. Resultant of Two Concurrent Forces. If two forces having the same point of application act on a body, there is some single force that might be applied at the same point to produce the same effect. This single force is called the resultant of the two forces, and is found as follows: construct upon the arrows representing the forces a parallelogram and draw the diagonal from the point of application. This diagonal represents the resultant of the two forces in magnitude and direction

(Art. 8). Thus, if P1 and P2 (Fig. 1) are the forces, then R is the resultant.

Algebraically R = √ P12 + P22 + 2 P1P2 cos AOB.

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12. Resolution of Force. We have just seen how two concurrent forces may be replaced by a single force called their resultant. In a similar way a single force may be resolved into two forces. These forces are the sides of a parallelogram of which the single force is a diagonal. It is clear, then, that there are an infinite number of components into which a single resultant may be resolved. It is necessary, therefore, in speaking of the components of a force, to state specifically which are intended. It will be seen in problems that follow that the components most often used are at right angles to each other, and usually the vertical and horizontal components. In such a case the components are the projections of the force on the vertical and horizontal lines.

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13. Force Triangle. It follows directly from the parallelogram law of forces (Art. 11) that if we draw from any point a line parallel to and representing one of two concurrent forces, P2 say, and from the extremity of this line another line parallel to P1 and of the same length, then the remaining side of the triangle will be represented by R. This triangle is called the force triangle. In general, the resultant of two concurrent forces may be found by drawing

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lines parallel to the forces as above. The line nec to complete the triangle is the resultant, and its ar. always away from the point of application. The equa opposite of this resultant would be a single force would hold the two concurrent forces in equilibriu

14. Force Polygon. current, we may find their resultant by proceeding in a similar to that outlined above. Thus, let the forces be P2, P3, P4, etc. (Fig. 2), all passing through a point;

If more than two forces are

parallel to P2, from the extremity of this last line dr another equal and parallel to P, and proceed in the sa way for the other forces. The figure produced will be polygon whose sides are equal and parallel to the force The resultant will be given in magnitude, direction, an point of application by the line necessary to close t polygon. The arrow, representing the direction of t resultant, will always be away from the point of applic tion. (See Fig. 2.) If the polygon be closed, the syste of forces will be in equilibrium. The single force neces sary to produce equilibrium will, in any case, be equal an opposite to R. The student should construct force poly gons by taking the forces in different orders and checking the resultant in each case.

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By drawing the lines OA, OB, OC, etc., it is easy to see that OA represents the resultant of P1 and P2, that OB represents the resultant of OA and P, and so of P1, P and P, etc. That is, it is easy to see that the force polygon follows directly from the force triangle. By means of the force polygon it is easy to find graphically the resultant of any number of concurrent forces in a plane. The work, however, must be done accurately.

The student should show that the force polygon may be used for finding the resultant of concurring forces in space, by considering two forces at a time. The force polygon in this case is called a twisted polygon.

15. Transmissibility of Forces. It is a matter of experience that the point of application of a force may be changed to any point along its line of action without changing the effect of the force upon the rigid body. This, of course, is on the assumption that all the force is transmitted to the body. The law may be stated as follows: The point of application of a force may be transferred anywhere along its line of action without changing its effect upon the body upon which it acts.

CHAPTER II

CONCURRENT FORCES

16. Concurrent Forces in a Plane. It will often be convenient to consider forces as acting on a material point; this is equivalent to considering the body without

weight and simply a point. If a material

point (0) (Fig. 3) be acted up

on by a number of forces in a plane, P1, P2, P3, P4 etc., each one mak

ing angles α1, Ag, aga etc., respectively,

with the posi

tive x-axis, it

is desirable to find the resultant of all of them in magnitude and direction; that is, the single ideal force that could produce the same effect as the system of forces.

Each force P may be resolved into components along the x- and y-axes, giving P cos a along the x-axis, and P

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