thread it will raise a weight more easily than one with a triangular thread: but in most if not all screws the friction is equal to the power. 5. In the inclined plane the friction varies according as the body rolls or slides; the friction in the latter case being far the greatest. 6. In the wedge the friction is at least equal to the power, since the wedge retains any position it is driven into.

29. A. The memoir of the same philosopher on the friction of pivots, is inserted among the Memoirs of the Paris Academy for the year 1790. Though it has been so long published, it is scarcely known even in France: yet as the experiments described are very interesting, and furnish some important results, it will be right to give an account of them.

Bodies which are made to turn upon pivots are usually suspended by means of a cheek, socket, or collar, of very hard matter. The collar has its cavity of a conic form, and terminated at its summit by a little concave segment, whose radius of curvature is very small. The point of the pivot which is sustained by this collar forms at its summit a little convex surface, whose radius of curvature should be still smaller than that of the extremity of the cheek. Experience evinces that the curvature of the bottom of the socket is irregular, and that the friction of a collar of agate on which a pivot turns, is frequently five or six times more considerable than the momentum of friction of a well-polished plane of agate on which the same pivot

turns.

These considerations induced M. Coulomb to employ in the course of his experiments, not a cheek or a socket, but a wellpolished plane, to support the body on the point of a pivot. To prevent the body from sliding he took care that its centre of gravity should be very low, with respect to the point of suspension he then made the body to whirl or spin about its pivot, by impressing upon it a rotatory motion. By means of a seconds watch, he observed exactly the time employed by the body in making the first four or five turns, and he thence deduced easily a mean turn to determine the primitive velocity: after this he counted the number of turns which the body made before it stopped.

Coulomb took a glass bell of 48 lines in diameter and 60 lines in height, which weighed 5 ounces. He placed it on the point of a pivot; and after giving it successive degrees of velocity about that pivot, he observed very exactly the time that it employed to make the first turn, which gave him for the mean velocity that which answered to the half of such first turn. He then estimated the number of turns made by the bell before it stopped: the results were as below

1st Trial. The bell made one turn in 4", and came to rest after 34

turns.

2d Trial. The bell made one turn in 61", and stopped after 14 turns.

3d Trial. The bell made one turn in 11", and stopped after

4 turns.

a

Now if b denote the primitive velocity, x the space described between the commencement and the end of the motion, a the constant momentum of the retarding force; the sum of the products of every particle, by the square of its distance r from the axis of rotation, divided by the quantity a measuring the distance from the axis of rotation to the point whose primitive velocity is b, it is easy to find the following analytical expression for the constant momentum of the vis retardatrix, viz.

But, because in the three preceding trials, the same bell was employed, the quantity is the same: must therefore be

a

X

a constant quantity if A be constant, and reciprocally. But in each trial there was reckoned the time employed by the apparatus in performing an entire revolution. The mean velocity, or the velocity due to the half of each first revolution, will, therefore, be measured by the circumference run over. The space described up to the end of the motion, will be measured by the number of turn run through from the instant where the mean velocity was determined until the end of the motion. Thus by computing from the data furnished by the three trials, we may form the following table:

1st Trial. 1 turn in 4", stops at 34 turns, whence results

X

This experiment, then, shews unequivocally that the quantity and consequently the quantity A which expresses the momentum of friction, are constant quantities, whatever be the primitive degree of velocity; and that, consequently, the velocity has not any influence upon the resistance due to the friction of pivots, which from this experiment is necessarily proportional to a function of the pressure.

When this experiment is made in a vacuum, a much less heavy body may be employed, and of any form whatever, and the same result will be obtained.

:

In other experiments Coulomb bent a brass wire of 9 inches in length; the parallel branches were 24 lines distant from one another; the part of the wire curved in the form of a semicircle which joined the two branches was about 3 inches long; and the two vertical and parallel branches were also each 3 inches long. To the extremity of each vertical branch was attached by means of wax a piece of metal, and there was fixed, in like manner, in the middle of the concave part of the wire, to serve for the cheek or bush, a small well-polished plane of different substances on which the friction of the point of the pivot was to be determined finally, there was fixed to the summit of a support a little needle of tempered steel, and whose point it was necessary to render more or less fine, rounded, or obtuse, according to the nature of the cheeks, and to the pressure which they were to experience. The extremity of the needle first used by Coulomb, appeared, when examined by a microscope, to form a conic angle of 18 or 20 degrees. The friction of this needle against well-polished planes of granite, agate, rock crystal, glass, and tempered steel respectively, was tried; and the result, taking in each experiment the mean quantity represented by — (a quantity which was always found to vary between very narrow limits), gave the momentum of friction of the point of the needle against the planes of granite, agate, &c. respectively, in the ratio of the fractions T i7, 784, 677, 487 so that the momentum of friction of the plane of granite being represented by unity, we shall have for the momentum of the friction of rotation relative to the other substances as below: friction of granite, 1; of agate, 1.214; of rock crystal, 1313; of glass, 1777; of steel, 2.257.

2

X

b

Coulomb likewise employed himself during these experiments, in determining the more or less acute form which should be given to the points of the pivots. To this end he caused to be successively rounded into cones of greater or less acuteness, the extremity of a steel needle, that it might thence appear whether the change of figure had any influence upon the friction. Thus he found that, under a certain charge, the point of the pivot being shaped to 45 degrees, the quantity granite, ; agate, '; glass, T; tempered steel, zʊʊʊ• Coulomb then gave to the point a more acute form, so that the angle of the cone which terminated it could not be more than 6 or 7 degrees; and he found, still retaining the same charge or pressure as before, that the quantity was, for agate, 'ʊ; glass, ; tempered steel, ʊ•

X

X

was, for

ba

which varies as that

Comparing from these, and other experiments, the momentum of friction of rotation of the point of different pivots against à plane of agate, he found that the quantity momentum, was, for a pivot of 45°, a pivot of 6, Tor.

; a pivot of 15°, τ2005

After this Coulomb varied the charge in his experiments, and determined the relative momentum of friction of pivots under different pressures. pressures. But without going further into detail, we may give the following as the principal deductions from the whole.

1. That the friction of pivots is independent of the velocities, being merely as a function of the pressure.

2. That the friction of granite is less than that of glass. 3. That the figure of the point of the pivot, as to acuteness, affects the quantity of friction; in such manner that when we cause to whirl upon the point of a needle, a body weighing more than 5 or 6 drams, the most advantageous angle for that point appeared to be from 30° to 45°; under a less pressure, the angle might be progressively diminished, without the friction being perceptibly augmented: it may even without great inconveni ence be reduced to 10° or 12° with good steel, when the charge does not exceed 100 grains: an important consideration in the suspension of light bodies upon cheeks or sockets.

These rules may be useful to the makers of chronometers. 30. Since cords and ropes are not perfectly flexible, it becomes necessary in estimating the advantages of pulleys, capstans, &c. to make some allowance for this want of flexibility: in this case we may have recourse to a theory which is far more satisfactory than any which has yet been invented with regard to friction, and which accords far better with experiment. The most useful formulæ may be deduced in a very small compass. Thus, let AC=CB=r, the radius of a pulley (fig. 3. pl. I.) and two weights w and g in equilibrio: if w should prevail, it is obvious that the cord Do becomes in the upper part bent so as to fit to the groove of the pulley, and in the lower part bent inwards so as to fall into the vertical bw: if the cord be tolerably flexible, the curving is pretty regular from в almost down to w: but if the cord be very rigid, BEW and ADO are found to be nearly straight lines, but neither of them vertical; the weight a being found to hang vertically below some point as a, making ca greater than CA, and the weight w hanging below some point b where ch is less than CB. So that as the arm of the lever at which one of the forces act is become greater, and that of the other less than r, the condition of equilibrium is no longer w= 9:

B

When the cord is only moderately rigid, as in most practical cases, the distance вb is always found so extremely small that it may be safely neglected in the discussion; that is, we need in such cases pay no regard to the want of flexibility in the part BEW corresponding to the weight w which is supposed to prevail; but merely enquire into that of the part A DO by which the other weight is suspended. Hence, if we put Aa=q, the condition of equilibrium will be expressed thus:

wr = q (r + q).

From this it results, that if w-o be the magnitude by which we should augment the power, that it may be on the point of prevailing; and if we have regard to the stiffness of the cord, this magnitude will be w-9=9.2. Consequently, to introduce the consideration of the stiffness of the cord employed in a machine, we have only to suppose that the arm of the lever at which the resistance acts is greater than it really is, by a determinate quantity q

It remains, then, to ascertain this quantity q: in order to which, it may be observed that a cord resists, on two accounts, the efforts which are made to bend it. The first is due to the tension of the cord, and is proportional to it, it will therefore bebo; the second is due to its warping or twisting, and we may represent by a the force employed to overcome it. Here a and bare, as is manifest, variable coefficients. Thus, for one and the same cord a + bo may represent the force required to bend it: but, if the cord be changed, the diameter d will be different, and we may conclude that, cæteris paribus, the force which must be employed will be proportional to a certain power n of d; for the force necessary to bend a cord will increase with its diameter: this power will decrease on the contrary with the radius r of the pulley; therefore (a+b) may represent the force necessary to overcome the stiffness of the cord; n being as yet an indeterminate quantity. This value being the augmentation which must be given to the force or weight w that it may be on the point of prevailing over the resistance Q, must, from what is before shewn, be equal to o. Thus we have

d"

dn

d" (a+bq) = qq, or q= = (a+bq). . (a).

This equation, it is true, is only furnished by general considerations, and not by a rigorous investigation: it contains, moreover, the unknown coefficients n, a, and b, varying for different cords. But there is a simple method of finding these coefficients, and of assuring ourselves that the expression is sufficiently exact in practice.