takes place) depend sensibly upon the area of the surfaces in contact. This, which is called Statical Friction, is thus capable of opposing a tangential resistance to motion which may be of any requisite amount up to μR; where R is the whole normal pressure between the bodies; and (which depends mainly upon the nature of the surfaces in contact) is the co-efficient of Statical Friction. This co-efficient varies greatly with the circumstances, being in some cases as low as 0.03, in others as high as 0.80. Later we shall give a table of its values. Where the applied forces are insufficient to produce motion, the whole amount of statical friction is not called into play; its amount then just reaches what is sufficient to equilibrate the other forces, and its direction is the opposite of that in which their resultant tends to produce motion. When the statical friction has been overcome, and sliding is produced, experiment shows that a force of friction continues to act, opposing the motion, sensibly proportional to the normal pressure, and independent of the velocity. But for the same two bodies the co-efficient of Kinetic Friction is less than that of Statical Friction, and is approximately the same whatever be the rate of

motion.

405. When among the forces acting in any case of equilibrium, there are frictions of solids on solids, the circumstances would not be altered by doing away with all friction, and replacing its forces by forces of mutual action supposed to remain unchanged by any infinitely small relative motions of the parts between which they act. By this artifice all such cases may be brought under the general principle of Lagrange (§ 254).

406. In the following chapters on Abstract Dynamics we will confine ourselves strictly to such portions of this extensive subject as are likely to be useful to us in the rest of the work, or are of sufficient importance of themselves to warrant their introduction-except in special cases where results, more curious than useful, are given to show the nature of former applications of the methods, or to exhibit special methods of investigation adapted to the difficulties of peculiar problems. For a general view of the subject as a purely analytical problem, the reader is referred to special mathematical treatises, such as those of Poisson, Delaunay, Duhamel, Todhunter, Tait and Steele, Griffin, etc. From these little is to be learned save dexterity in the solution of problems which are in general of no great physical interest the objects of these treatises being professedly the mathematical analysis of the subject; while in the present work we are engaged specially with those questions which best illustrate physical principles.

CHAPTER VI.

STATICS OF A PARTICLE.-ATTRACTION.

407. WE naturally divide Statics into two parts-the equilibrium of a Particle, and that of a rigid or elastic Body or System of Particles whether solid or fluid. The second law of motion suffices for one part-for the other, the third, and its consequences pointed out by Newton, are necessary. In the succeeding sections we shall dispose of the first of these parts, and the rest of this chapter will be devoted to a digression on the important subject of Attraction.

408. By § 221, forces acting at the same point, or on the same material particle, are to be compounded by the same laws as velocities. Therefore the sum of their resolved parts in any direction must vanish if there is equilibrium; whence the necessary and sufficient conditions.

They follow also directly from Newton's statement with regard to work, if we suppose the particle to have any velocity, constant in direction and magnitude (and § 211, this is the most general supposition we can make, since absolute rest has for us no meaning). For the work done in any time is the product of the displacement during that time into the algebraic sum of the effective components of the applied forces, and there is no change of kinetic energy. Hence this sum must vanish for every direction. Practically, as any displacement may be resolved into three, in any three directions not coplanar, the vanishing of the work for any one such set of three suffices for the criterion. But, in general, it is convenient to assume them in directions at right angles to each other.

Hence, for the equilibrium of a material particle, it is necessary, and sufficient, that the (algebraic) sums of the applied forces, resolved in any one set of three rectangular directions, should vanish.

409. We proceed to give a detailed exposition of the results which follow from the first clause of § 408. For three forces only we have the following statement.

The resultant of two forces, acting on a material point, is repre

sented in direction and magnitude by the diagonal, through that point, of the parallelogram described upon lines representing the forces.

410. Parallelogram of forces stated symmetrically as to the three forces concerned, usually called the Triangle of Forces. If the lines representing three forces acting on a material point be equal and parallel to the sides of a triangle, and in directions similar to those of the three sides when taken in order round the triangle, the three forces are in equilibrium.

Let GEF be a triangle, and let MA, MB, MC, be respectively equal and parallel to the three sides EF, FG, GE of this triangle, and in directions similar to the consecutive directions of these sides in order. The point M is in equilibrium.

B

AG

M

E

H

411. [True Triangle of Forces. Let three forces act in consecutive directions round a triangle, DEF, and be represented respectively by its sides: they are not in equilibrium, but are equivalent to a couple. To prove this, through D draw DH, equal and parallel to EF, and in it introduce a pair of balancing forces, each equal to EF. Of the five forces, three, DE, DH and FD, are in equilibrium, and may be removed; and there are then left two forces, EF and HD, equal, parallel, and

D

in dissimilar directions, which constitute a couple.]

E

412. To find the resultant of any number of forces in lines through one point, not necessarily in one plane

Let MA,, MA, MA, MA, represent four forces acting on M, in one plane; required their resultant.

Find by the parallelogram of forces, the resultant of two of the forces, MA, and MA,. It will be represented by MD. Then similarly, find MD", the resultant of MD (the first subsidiary resultant), and MA,, the third force. Lastly, find MD"", the resultant of MD" and MA, MD"" represents the resultant of the given forces.

D"

D"

D'

Аз

A2

M

A4

Thus, by successive applications of the fundamental proposition, the resultant of any number of forces in lines through one point can be found.

413. In executing this construction, it is not necessary to describe

the successive parallelograms, or even to draw their diagonals. It is

Ꭺ

D

A2

D"

D'

enough to draw through the given point a line equal and parallel to the representative of any one of the forces; through the point thus arrived at, to draw a line equal and parallel to the representative of another of the forces, and so on till all the forces have been taken into account. In this way we get such a diagram as the annexed.

The several given forces may be taken in any order, in the construction just described. The resultant arrived at is necessarily the same, whatever be the order in which we choose to take them, as we may easily verify by elementary geometry.

A5

M

D

In the fig. the order is MA,, MA5, MA2, MA, MA ̧.

414. If, by drawing lines equal and parallel to the representatives of the forces, a closed figure is got, that is, if the line last drawn leads us back to the point from which we started, the forces are in equilibrium. If, on the other hand, the figure is not closed (§ 413), the resultant is obtained by drawing a line from the starting-point to the point finally reached; (from M to D): and a force represented by DM will equilibrate the system.

D

D"

415. Hence, in general, a set of forces represented by lines equal and parallel to the sides of a complete polygon, are in equilibrium, provided they act in lines through one point, in directions similar to the directions followed in going round the polygon in one way.

416. Polygon of Forces. The construction we have just considered, is sometimes called the polygon of forces; but the true polygon of forces, as we shall call it, is something quite different. In it the forces are actually along the sides of a polygon, and represented by them in magnitude. Such a system must clearly have a turning tendency, and it may be demonstrated to be reducible to one couple.

417. In the preceding sections we have explained the principle involved in finding the resultant of any number of forces. We have now to exhibit a method, more easy than the parallelogram of forces affords, for working it out in actual cases, and especially for obtaining a convenient specification of the resultant. The instrument employed for this purpose is Trigonometry.

418. A distinction may first be pointed out between two classes of problems, direct and inverse. Direct problems are those in which the resultant of forces is to be found; inverse, those in which com

ponents of a force are to be found. The former class is fixed and determinate; the latter is quite indefinite, without limitations to be stated for each problem. A system of forces can produce only one effect; but an infinite number of systems can be obtained, which shall produce the same effect as one force. The problem, therefore,

of finding components must be, in some way or other, limited. This may be done by giving the lines along which the components are to act. To find the components of a given force, in any three given directions, is, in general, as we shall see, a perfectly determinate problem.

Finding resultants is called Composition of Forces.

Finding components is called Resolution of Forces. 419. Composition of Forces.

Required in position and magnitude the resultant of two given

forces acting in given lines on a material point.

Let MA, MB represent two forces, P and Q, acting on a material point M. Let the angle BMA be denoted by . Required the magnitude of the resultant, and its inclination to the line of either force.

M

B

Q

R

A

P

Let R denote the magnitude of the resultant; let a denote the angle DMA, at which its line MD is inclined to MA, the line of the first force P; and let ẞ denote the angle DMB, at which it is inclined to MB, the direction of the force Q.

Given P, Q, and required R, and a or ẞ. We have

MD2 = MA2+ MB2-2 MA.MB x cos MAD.

Hence, according to our present notation,

R2 = P2+Q2-2PQ cos (180°),

or

R2 = P2+Q2+2 PQ cos .

Hence

R = (P2+Q2+2PQ cost).

(1)

To determine a and ẞ after the resultant has been found; we have

420. These formulae are useful for many applications; but they have the inconvenience that there may be ambiguity as to the angle, whether it is to be acute or obtuse, which is to be taken when either sina or sinẞ has been calculated. If is acute, both a and ẞ are acute, and there is no ambiguity. If is obtuse, one of the two