51. The following is an exercise on the parallelogram of forces.

VI. Assuming the truth of the parallelogram of forces (Art. 18) for the magnitude of the resultant, prove it also for the direction of the resultant.

Let three forces P, Q, R acting in one plane at a point A be in equilibrium, and let them be

represented by AB, AC, AE respectively.

Complete the parallelogram BC; then by the assumption the diagonal AD represents the resultant of P, Q in magnitude:-and since any one of the three forces P, Q, R is equal in magnitude to the resultant of the other two, it follows that AE= AD.

B

Complete the parallelogram BE; then AF represents the resultant of P, R in magnitude, and therefore AF = AC.

Hence BF, AD are equal, since they are each equal to AE,

and AF, BD are equal, since they are each equal

to AC,

that is, the opposite sides of the quadrilateral AFBD are respectively equal to each other, and therefore AFBD is a parallelogram.

Hence AD, AE being each parallel to BF are in the same straight line;—which proves the parallelogram of forces for the direction of the resultant.

CHAPTER III.

OF FRICTION.

52. WHEN a heavy body rests on a plane horizontal surface, on a table for example, and we wish to make it slide along the surface, we encounter a resistance to this motion; there exists between the particles of the body and the table an adhesion which resists their separation, and this adhesion is only overcome by applying to the body a force of traction sufficiently great. This adhesive force is called friction, and the magnitude of the force which is necessary to overcome the resistance to motion will be a measure of the friction.

More generally, when one surface presses against another, if the direction of this pressure be not normal to the surfaces in contact, there will be a tendency of one surface to rub or slide over the other; and no sliding motion will ensue, unless the resolved part of the pressure along the surface be sufficient to overcome the friction. When a body is just on the point of sliding, it is said to be in a state bordering on motion, and the greatest amount of friction which the surfaces can exert is then in operation. In other cases no more friction is called into action than is just sufficient to balance the part of the pressure resolved along the surface in contact.

In this point of view, friction may be called a self-adjusting force, since it adapts itself to the requirements of each particular case; no more being called into operation than is just necessary to prevent motion.

53. If R be the normal pressure between two surfaces in contact, F the friction when the bodies are just on the point of sliding over each other, i.e. the maximum friction which

F

the substances can exercise, the ratio is called the coefficient

R

of friction, and is commonly designated by μ:

so that F=μR.

If in any particular case the full amount of friction which the substances can exert is not called into action, the amount of friction which is actually in operation is one of the unknown forces which it is the object of the problem to determine.

54. The results of careful experiments made with the object of determining the laws of friction are thus given by Coulomb, and M. Morin: viz.

(i) When the substances in contact remain the same, the friction varies as the pressure; i.e. μ is the same for the same substances, but will vary for different substances. When the pressure is very great indeed, it is found that the friction is a little less than this law would give.

(ii) So long as the normal pressure between the surfaces in contact remains the same, the whole amount of friction is independent of the extent of surface in contact.

These two laws are true when the body is in a state bordering on motion, and also when actually in motion; only it is to be remarked that in the latter case the magnitude of the friction is much less than in the former. If we call the friction in the former case statical, and that in the latter dynamical, we may express the above by saying that the coefficients of dynamical and statical friction are severally constant

for the same substances, but that the dynamical is less than the statical.

It is also found:

(iii) That the friction is independent of the velocity when the body is in motion.

55. The friction between two bodies will generally be diminished by smearing them with some unctuous substance, as oil, &c., and the friction when they are on the point of moving, or what we may call the friction at starting, is pretty nearly the same as during motion when the bodies are made of hard material, like stone or metal. But in the case of compressible substances like wood, the friction at starting is very considerably greater than during motion. When two bodies are placed one upon the other, one of them at least being compressible, the amount of friction at starting will partly depend upon the length of time they have been in contact. For wood sliding upon wood, the maximum friction is attained after a contact of a few minutes; but for wood upon metal it requires a much longer time, frequently several days for the friction to attain its maximum: but when it has attained this, the friction at starting is not altered by any continued duration of contact.

Further, it is found that rolling friction is much less than sliding friction: for example, when a cylinder rolls on a plane, or a cylindrical axis turns within a hollow socket (when there is simply a line in contact and not a finite area), the amount of friction is much less than would be given by the above laws (i) and (ii), for the same amount of pressure.

The fact that rolling friction is much less than sliding friction is taken advantage of in various contrivances for faci

litating the transport of heavy bodies; thus for instance, heavy blocks of stone or other material are often transported by placing them on a platform beneath which rollers are placed: the wheels of carriages are examples of the same principle,—the most delicate application of which perhaps is that of friction wheels, such as those employed in Atwood's Machine (see Dynamics, Art. 82).

56. The values of μ for different substances have been determined by experiment, and arranged in tables; the following may be taken as approximate results in many cases for friction at starting:

When a cylinder of wood rolls upon wood so that there

1

is a single line of contact only, μ=12 ; -when the surface in

contact is a physical point the statical friction is inconsiderable.

57. To find the coefficient of friction between two substances practically.

Let AB be the plane surface of one substance, upon which