Names of the wood used for axles. 7. Do. the coating wiped, and the surfaces remaining greasy, 8. Axis of boxwood, and box of elm, coated with tallow, 9. Do. the coating wiped off, and the surfaces remaining greasy, 10. Axis of iron, and box of lignum vitæ, the coating wiped off, and } the pulley turned for some time, Ratio of friction to pressure. 1 14.3 1 28.6 1 20 20 In these experiments the velocity did not appear to influence the friction unless in the first instants of rest; and in all cases the friction was least when the surfaces were merely greasy, and not coated with tallow. On the Friction and Rigidity of Ropes. When a rope passes over a cylinder, or over the groove of a pulley, whose axis is horizontal, and has a weight suspended at each end of it, then it is obvious that if the rope were destitute of rigidity, or perfectly flexible, the axis of the ropes on each side would be perfectly parallel and vertical. But as every rope is stiff or rigid, the two branches of it that suspend the weights will deviate from the perpendicular. If one of the ends of the rope is fixed in such a manner that the first branch of the rope is in a vertical direction, and if we suspend, at the other end of the rope, a weight W, which will stretch the second branch of the rope to such a degree as to force it into a vertical direction, and to bring it in contact with a semicircumference of the pulley, the weight W may be taken as the measure of the rigidity of the rope. If the rope, instead of being treated in this manner, has two equal weights, W, W, suspended at each extremity, and if we add to one of the ends an additional weight w, capable of destroying the equilibrium of the equal weights, W, W, this weight may be taken as a measure of the friction of the rope. A small portion indeed of the weight w is employed in overcoming the rigidity of the rope; but it is so small, when compared with w, that it may be safely neglected. The first experiments that appear to have been made on the rigidity of ropes, were those of Amontons, who contrived an ingenious apparatus for this purpose. He has published, in the Memoirs of the Academy of Sciences for 1699, a table of the forces required to bend ropes, founded on the supposition that the difficulty of bending a rope of the same thickness, and loaded with the same weight, "decreases when the diameter of the roller or pulley increases, but not so much as that diameter increases." Another series of experiments was afterwards made by Desaguliers, (See Course of Natural Philosophy, vol. i, p. 243, 244, 245, &c.) who published a table," shewing what forces were required to bend ropes of different diameters, stretched by different weights round rollers of different bignesses." The general result of these experiments was, that the difficulty of bending a rope round a roller, is ceteris paribus inversely as the diameter of the roller. The experiments on the rigidity and friction of ropes made by Coulomb, were performed, both with the apparatus used by Amontons, and by another of his own. It consists of two tressels 6 feet high, on which are laid two pieces of square wood. On these two pieces are fixed two rulers of oak, well planed, and polished with fish-skin. With two cylinders of lignum vitæ, one 6 inches in diameter, and the other 2 inches; and, with several cylinders of elm, from 2 to 12 inches in diameter, the apparatus was ready for the experiments. In order to ascertain the friction of the rollers, they were laid on the planks, and a weight of 50lbs. was suspended on each side of the roller, with very fine and flexible pack-thread; and the weights could be increased in any degree by laying additional weights of 50lbs. by different threads, so as to give any required pressure to the rollers. By adding a counterweight on each side of the roller alternately, till they received a motion barely sensible, Coulomb ascertained the friction of the rollers. The following were the results with rollers of lignum vitæ : Hence it follows, that the friction of cylinders rolling upon horizontal planes are directly as the pressures, and inversely as the diameters of the rollers. Coulomb found that greasing the ropes did not diminish the friction sensibly. He found also, that rollers of elm had a friction about two-fifths greater than lignum vitæ; and that under small pressures the friction was rather greater than would result from its being proportional to the pressure. The rigidity of ropes was measured in the way already described, and the following results were obtained. The rollers varied from 2 to 12 inches in diameter, and the pulley ropes were used. Threads in a yarn. Threads in a strand. Circumference. 12 lines. 15 5 20 30 10 28 Weight of a foot. 4 drs. 121 24 After making a series of experiments in the case of motions nearly insensible, Coulomb proceeded to ascertain the effect produced by the rigidity of the cords as changed by the velocity. With this view, he took a pulley, and a box of copper, and an axis of iron, coated with tallow. The pulley was 144 lines in diameter, and the axis 20 lines, and the cord was No. 3, the same as that used in the above experiment. The weights were made to run above a distance of 6 feet, and the times of describing the first three and the last three feet, were measured by a half-seconds pendulum. The following are the general results of both sets of experi ments: 1. The rigidity of ropes increases, the more that the fibres of which they are composed are twisted. 2. The rigidity of ropes increases in the duplicate ratio of their diameters. According to Amontons and Desaguliers, the rigidity increases in the simple ratio of the diameters of the ropes; but this probably arose from the flexibility of the ropes which they employed. 3. The rigidity of ropes is in the simple and direct ratio of their tension. 4. The rigidity of ropes is in the inverse ratio of the diameters of the cylinders round which they are coiled. This result was obtained also by Desaguliers. 5. In general, the rigidity of ropes is directly as their tensions and the squares of their diameters, and inversely as the diameters of the cylinders round which they are coiled. 6. The rigidity of ropes increases so little with the velocity of the machine, that it need not be taken into the account when computing the effects of machines. 7. The rigidity of small ropes is diminished when penetrated with moisture; but when the ropes are thick, their rigidity is increased. 8. The rigidity of ropes is increased, and their strength di minished, when they are covered with pitch; but when ropes of this kind are alternately immersed in the sea and exposed to the air, they last longer than when they are not pitched. This increase of rigidity, however, is not so perceptible in small ropes as in those whichare pretty thick. 9. The rigidity of ropes covered with pitch is a sixth part greater during frost than in the middle of summer, but this increase of rigidity does not follow the ratio of their tensions. 10. The resistance to be overcome in bending a rope over a pulley or cylinder may be represented by a formula composed a Dn of two terms. The first term is a constant quantity inder pendent of the tension, a being a constant quantity determined by experiment, D" a power of the diameter D of the rope, and r the radius of the pulley or cylinder round which the rope is coiled. The second term of the formula is Tx bDn where T is the tension of the rope, b a constant quantity, and Dn and r the same as before. Hence the complete formula is a Dn r + Tx bDn Dn = xa+Tb. The exponent n of r the quantity D diminishes with the flexibility of the rope, but is generally equal to 1.7 or 1.8; or, as in No. 2, the rigidity is nearly in the duplicate ratio of the diameter of the rope. When the cord is much used, its flexibility is increased, and n becomes equal to 1.5 or 1.4.* On the Friction and Form of Pivots. The needles of compasses are generally suspended upon a pivot, by means of caps of agate, or other hard substances. The cap has a conical form, terminated above with a small concave summit, whose radius of curvature is very small. The pivots themselves are commonly of tempered steel, but frequently reduced to the state of a spring. The point of the pivot which supports the cap is a small curve surface, whose radius of curvature is smaller than that of the summit of the cap. Coulomb generally found, that even when every care was • A drawing of the apparatus employed, and tabular views of all the circumstances and results of the experiments, will be found in the Edinburgh Encyclopædia, Art, Mechanics, vol. xiii, p. 599, 600. cap was taken by the artist, the curvature of the summit of the very irregular, and that the friction of an agate cap, turning upon the point of a pivot, was often five or six times greater than the momentum of friction of a highly polished agate plane turning on the same pivot. In his experiments on pivots, Coulomb supported the body by a highly polished plane in place of a cap, and having given it a rotatory motion, he noted the time employed in making the four or five first turns, from a mean of which he obtained the primitive velocity; and he next counted the number of turns which it made before it stopped. The revolving body is obviously brought to rest by the friction of the point of the pivot, and also by the resistance of the air; but in order to get rid of this last resistance, Coulomb gave the body the form of a glass receiver, and found, that when the velocity was not great, and when the receiver weighed 5 or 6 gros (a gros is the eighth of an ounce), the resistance of the air bore no sensible ratio to that of the friction. In order to render the results more certain, he made several of the experiments in vacuo by an apparatus protected from currents of air by being placed under a receiver. With the apparatus, which consisted of a glass receiver 4 inches in diameter, 5 inches high, and weighing 5 ounces, Coulomb obtained the following results : By calculating the friction from a formula which he had previously given in the Memoirs of the Academy for 1779, p. 451, Coulomb found that the friction of pivots is independent of the velocity, and is therefore necessarily proportional to some function of the pressure. In order to examine the friction of pivots, when they support planes of different materials, M. Coulomb made the following experiments. The angle of the summit of the pivot which supported the planes was about 18 or 20 degrees. * The reader will find this formula and a drawing of the apparatus in the Edinburgh Encyclopædia, Art. Mechanics, vol. xiii, p. 601. |