4. Friction is a uniformly retarding force.1

Although friction is a deteriorating force to bodies in motion, yet when a body is required to be kept at rest, it is an advantage. Without it the wedge would be a useless machine, for it would recoil after each blow; and nails driven into a board would not hold. By friction the ropes of a ship are fastened by merely twisting them round a belaying-pin. Indeed, without friction, the aspect of things in general would be very much changed; horses could not draw loads, nor could men or animals walk, and they would helplessly slide to the bottom of any slope upon which they were placed.

II. Rolling Friction.-When a body rolls over the surface of another, the particles, instead of being drawn on over each other, are merely laid down and lifted up. As the surface of one of the bodies, at least, must be bounded by a curved line, which can only touch another line at a line or a single point, the surfaces of contact are reduced to a minimum.

If the rolling body be a cylinder, the points of contact will be a line parallel to the axis of the cylinder; but if the body be a globe, the area of contact will be a point.

Cylinders. To ascertain the ratio between the friction and pressure of rolling bodies, we will adopt the angle of the inclined plane as our test. Thus, in fig. 104, let AB be the inclined plane, upon which the

1 That is to say, that the amount of force expended in overcoming friction is proportional to the velocity, since a constant force must be exerted with a varying velocity.

Hence, whatever be the ratio between the force and friction of two moving surfaces of any machine moving with a given velocity, when the velocity is doubled,—the pressure on the surfaces remaining the same, the ratio between the force and friction will also be doubled.

It also follows, that the amount of force lost by friction will in every case be proportional to the space passed over by the surfaces in contact, without regard to time.

Friction, therefore, is always the same fraction of the pressure, however great (within certain limits), and however small that pressure may be.

cylinder H is placed, of the same weight as the body used in the former experiments. By the elevation of

the screw, the angle of repose is found to be much smaller for equal weights when the surface of one of the bodies is round, than where both the surfaces in contact are flat. In this case, suppose H to be of twelve pounds weight. The friction is to the pressure as BC to A C.

Again; let K (fig. 105) be a cylinder of equal weight to H, but of double its diameter. The friction will again be to the pressure as B'C' to A'C'. But A C being common to the two planes, it is shown that the friction of a cylinder of double diameter but equal weight is to that of the first cylinder as B'C' to B C.

The equation to express the force necessary to overcome this friction when acting at the circumference of the roller, may be represented thus:

Let P-pressure, R=radius, and f= coefficient of friction. Then

F=√2R

In a third case, let M (fig. 106) be a cylinder of three inches diameter and six pounds weight; B C represents its friction, and A C its pressure.

Then, let N (fig. 107) be a cylinder of five inches diameter and eight pounds weight, B'C' represents its friction and A' C' its pressure. By comparing the

lengths of AC with A' C', and B C with B'C', it will be found that the friction of the three-inch cylinder will be to that of the five inch-cylinder, as 5×6: 3×8, or as 5: 4.

Axioms.-Wherefore, in experiments with a body rolling over the surface of another, it is found that

1.-With cylinders of the same substances and equal diameters, the friction is as the pressure.

2. With cylinders of equal pressures and the same substances, but of different diameters, the friction is inversely as the diameters.

3.-With cylinders of the same substances but different diameters and pressures, the friction is directly as the pressures and inversely as the diameters.

Globes. The angle of repose of a globe is found to be less than that of any other shaped body. Wherefore, the friction of a globe is the minimum of friction with an equal pressure.

The application of unguents to the surfaces of either of the bodies in contact does not diminish this kind of friction in the least degree.

The following table, extracted from Morin's Notions Fondamentales de Mécanique, show some of the results of experiments on the friction of various bodies:

Angle of

Repose.

311°

0.62

2810

0.54

perpendicular.

Oak on Elm, ,, parallel.

Elm on Oak,

99

Wood on Wood, dry.

2° to 111°

0.2 to 0.04

CHAPTER XV.

RIGIDITY OF CORDAGE.

IN estimating the mechanical efficiency of systems of pulleys, we have neglected the consideration of the stiffness of the fall, although force is necessary to bend the rope into an arc over the pulley, and again to straighten it after it has passed over.

In determining the amount of resistance due to rigidity, it is necessary to consider the forces applied to act along the centre of the rope, or with a leverage composed of the radius of the pulley and half the diameter of the rope. This is the effective radius of the sheave.

B

B

Fig. 108.

In fig. 108, D is a sheave over which the rope C D E passes, to the ends of which equal weights, A and B, are suspended.

The part of the rope CDE is bent into an arc, and has a tendency to retain that form by reason of its stiff

ness.

If a small additional weight be suspended from A, it will cause the sheave to rotate in the direction ED C. Owing to the rigidity of the rope, the cords will not become straightened immediately upon leaving the sheave, but will take the form shown by the dotted line. From A' and B' draw lines, A' G, B' I, perpendicular to G I, drawn through the centre of the pin at right angles to

L., rigidus, stiff.

the direction of the strain. It is evident that the weight A' is now acting with the leverage GH, and the weight B' is acting at the end of the lever H I. Owing to the stiffness of the rope, the leverage of the working part of the rope is diminished, while the loaded part has an increased leverage. If the force applied to A, multiplied by the distance GH, is not greater than the weight B multiplied by the length H I, no motion can ensue; so that unequal weights may be placed at opposite ends of a rope passing over a pulley, and the machine will still continue in equilibrium. It is on this account that while theoretically the pulley is by its action applicable to the purposes of a balance, it is practically of little use in

this direction.

The forces opposed to the mechanical efficiency of the pulley, may be classed under two heads-(a), those due to the rigidity of the rope; and (b), those relating to the weight upon the pulley. As the curvature of the rope over the sheave increases, so the resistance due to rigidity becomes greater; this resistance also increases faster than the radius of the rope.

In winding a rope off a drum upon which it has been coiled, the rigidity is not taken into account.

The table annexed is the result of Morin's calculations from the experiments of Coloumb :—