CHAPTER III.

ELEMENTARY STATICS.

23. Mechanics.-The science of Mechanics is that which treats of the motion and rest of bodies as produced by force. The words as produced by force' are added in order to exclude the science of pure motion or mechanism, which treats of the forms of machines, and in which machines are regarded merely as modifiers of motion. Into all questions which are properly mechanical the idea of force must enter.

Force may be defined to be any cause which puts a body in motion, or which tends to put a body in motion when its effect is hindered by some other cause. On this definition the following remark is to be made: Suppose a given weight is supported by a string passing over a pulley and fastened to a fixed point at the other end; next, suppose an equal weight to be supported by a man's hand; lastly, suppose an equal weight to be supported by the expansive pressure of a spring. Now, here we have three physical agents, viz. the reaction of the fixed point transmitted through the string, the muscular power of a man, and the elastic power of a spring, very different in many respects, but agreeing in their common capacity to support a given weight. They may clearly be regarded as equal, when viewed with reference to that capacity. In short, as in geometry, we regard all bodies as equal which can successively fill the same space, without any regard to their physical qualities, such as weight, colour, &c., so in mechanics we regard all forces as equal which will severally balance by direct opposition a given weight irrespectively of their physical origin. By the weight of

a body is meant the mutual attraction between the earth and that body; as this attraction has different amounts when the body is at different places, the weight of a body, when used as a standard of force, must be determined with reference to some assigned place. Thus :there is kept in the Exchequer office a piece of platinum called the standard pound (avoirdupois); the attraction of the earth on that body at London is a force of 1 lb., and any force which by direct opposition can support that body in London is also a force of 1 lb. If we suppose two forces each of 1 lb. to act in the same direction on a point and to be balanced by a single force, that force is one of 2 lbs.; and similarly a force of three, four, or more pounds can be defined.

24. Statics and Dynamics.—It follows, from the definition, that, in Mechanics, we can consider a force either as producing motion, or as concurring with others in producing rest. Accordingly, the science of mechanics is divided into two distinct though closely connected branches, viz. statics and dynamics. Of these, statics is that science which determines the conditions of the equilibrium of any body or system of bodies under the action of given forces. Dynamics is that science which determines the motion, or the change of motion, that ensues in a body or system of bodies subjected to the action of a force or forces that are not in equilibrium.

25. Determination of a Force. From what has already been said, it appears that the magnitude of any force is assigned by considering the weight it would just support if applied directly upward; in other words, we arrive at the magnitude of any force by comparing it with the most familiar and measurable of forces, viz. weight. A little consideration will show that the effect of a force in any case depends not only on its amount but also on its point of application, and the line along which

it acts.

P

FIG. 11.

B

We may say, therefore, in general terms, that a force is completely determined when we know (1) its magnitude, (2) its point of application, (3) its line of action, and (4) its direction along that line.* A lige is frequently said to represent a force; when this is the case, it must be drawn from the point of application of the force along the line of its action, and must contain as many units of length (say inches) as the force contains units of weight (say lbs.). It is of great importance that the

B

student should attend to all the conditions which must meet when a line correctly represents a force. Suppose a force of p lbs. (fig. 11) to act on a body at the point a ; if the force is a pull, as in the first figure, the line AB containing as many inches as P contains lbs. will represent the force; but if the force is a push, AB must be measured, as in the second figure.

26. Resultant and Components.-If we consider any forces that keep a body in equilibrium, it is plain that any one of them balances all the

others: thus, if three strings be knotted together at A, and be pulled by forces of P lbs., q lbs., and R lbs. respectively so adjusted as to balance one another, it is plainly a matter of indifference whether we

FIG. 12.

R

A

R'

consider that P balances Q and R, or that Q balances R and P, or that R balances P and Q. Let us consider that R

*The student must notice the distinction between the line of action and the direction of a force: e. g. in Fig. 14 (p. 51) P and Q act in the same direction along different lines.

E

balances P and Q; now R would of course balance a force R' exactly equal and opposite to itself; so that if we substitute R' for P and Q, or vice versa, P and Q for R', in either case R is balanced, and the force R' is equivalent to P and Q; under these circumstances, R' is called the resultant of P and Q; and P and Q are called the components of R'. Hence we may state generally,

Def. That force which is equivalent to any system of forces, is called their resultant.

Def. Those forces which form a system equivalent to a single force, are called its components.

27. Resultant of Forces acting along the same Straight Line.--If the forces act in the same direction the resultant must be their sum. If some act towards the right and some towards the left, the first set can be formed into a single force (P) acting towards the right, the second set can be formed into a single force (9) acting towards the left: the resultant of these two, and therefore of the original set of forces, will be equal to the difference between P and Q and will act in the direction of the greater. If the forces are in equilibrium the sum of those acting towards the right must equal the sum of those acting towards the left.

Ex. 164.-If three men pull on a rope to the right with forces of 31, 20, and 27 lbs. respectively, and are balanced by two men who pull with forces of 40 and p lbs. respectively, find P.

P

Ans. 38 lbs.

Ex. 165.-In the last example find the resultant of the 5 forces (1) if P = 30 lbs.; (2) if p=40 lbs. Ans. (1) 8 lbs. acting towards the right. (2) 2 lbs. acting towards the left.

Ex. 166.-There is a rope AB and men pull along it in the following manner: the first with a force of 50 lbs. towards A; the second with a force of 37 lbs. towards B; the third with a force of 35 lbs. towards ▲; the fourth with a force of 20 lbs. towards A; the fifth with a force of 54 lbs. towards B; the sixth with a force of 27 lbs. towards A; the seventh with a force of 52 lbs. towards B; the eighth with a force of 30 lbs. towards Determine the single force that must act along AB to balance them, and find whether it acts towards A or B. Ans. 41 lbs. acting towards A.

B.

Ex. 167.- In the last example suppose the second force to act towards A, find the resultant. Ans. 33 lbs. acting towards A.

FIG. 13.

R

B

28. The terms Reaction, Thrust, Strain, and Tension are of frequent occurrence in Mechanics. They may be most readily explained with reference to the equilibrium of two forces. Let AB (fig. 13) be a body urged by a force T against a fixed plane AC, and let the motion which T tends to communicate to the body be prevented by the fixed plane; that fixed plane must supply a force (R) which exactly balances T; and the body AB is really compressed between two forces R and T, of which the former is the Reaction of the fixed surface, and the latter the Thrust along A B. A Thrust and a Reaction compress or tend to compress the body on which they act. If, on the contrary, the body (AB) had been acted on by two equal opposite forces T and R tending to produce elongation, it is said to sustain a strain T. There is no essential difference between a strain and a tension; the former term is generally used when the body is inflexible, the latter when the body is flexible; thus, we speak of the strain on a tie beam, and the tension of a cord. One of the forces producing a strain or a tension may, of course, be a passive resistance like a reaction; thus if one end of a string is tied to a nail fast in a post, and the other end to a weight of 10 lbs., the string is stretched by two forces each of 10 lbs., viz. the weight and the reaction of the nail, and the string is said to sustain a tension of 10 lbs.

A

C

FIG. 14.

R

A

с

B

R'

29. Resultant of two Parallel Forces.First, let P and Q (fig. 14) be the two parallel forces acting in the same direction at the points