in mechanics, is a subject of great importance both to the practical engineer and to the speculative philosopher. It is therefore our duty to correct, in this Supplement, the mistakes into which we fell when treating of that subject in the Encyclopedia. What we have there taught of friction (see Mechanics Sect. II. § 8.) is taken from Ferguson; but it has been shewn by Mr Vince, that the experiments from which his conclusions were drawn were not properly instituted. That eminent mathematician and philosopher therefore entered upon the investigation of the subject anew, and endeavoured, by a set of experiments, to determine the following questions:
1. Whether friction be a uniformly retarding force? 2. The quantity of friction? 3. Whether the friction varies in proportion to the pressure or weight? 4. Whether the friction be the same on whichever of its surfaces a body moves?
1. With respect to the first of these questions, the author truly observes, that if friction be a uniform force, the difference between it and the given force of the moving power employed to overcome it must also be uniform; and that therefore the moving power, if it be a body descending by its own weight, must descend with a uniformly accelerated velocity, just as when there was no friction. The spaces described from the beginning of the motion will indeed be diminished in any given time on account of the friction; but still they must be to each other as the squares of the times employed. See Dynamics in this Supplement.
2. A plane was therefore adjusted parallel to the horizon, at the extremity of which was placed a pulley, which could be elevated or depressed, in order to render the string which connected the body and the moving force parallel to the plane. A scale accurately divided was placed by the side of the pulley perpendicular to the horizon, by the side of which the moving force descended; upon the scale was placed a moveable flag, which could be adjusted to the space through which the moving force descended in any given time; which time was measured by a well-regulated pendulum clock vibrating seconds. Everything being thus prepared, the following experiments were made to ascertain the law of friction.
3. Exp. 1. A body was placed upon the horizontal plane, and a moving force applied, which, from repeated trials, was found to descend 54 inches in 4"; for by the beat of the clock, and the sound of the moving force when it arrived at the flag, the space could be very accurately adjusted to the time: The flag was then removed to that point to which the moving force would descend in 3", upon supposition, that the spaces described by the moving power were as the squares of the times; and the space was found to agree very accurately with the time: the flag was then removed to that point to which the moving force ought to descend in 2", upon the same supposition, and the descent was found to agree exactly with the time: lastly, the flag was adjusted to that point to which the moving force ought to descend in 1", upon the same supposition, and the space was observed to agree with the time. Now, in order to find whether a difference in the time of descent could be observed by removing the flag a little above and below the positions which corresponded to the above times, the experiment was tried, and the descent was always found too soon in the former, and too late in the latter case; by which the author was assured, that the spaces first mentioned corresponded exactly to the times. And, for the greater certainty, each descent was repeated eight or ten times; and every caution used in this experiment was also made use of in all the following.
Exp. 2. A second body was laid upon the horizontal plane, and a moving force applied which descended 4 inches in 3"; the stage was then adjusted to the space corresponding to 2", upon supposition that the spaces descended through were as the squares of the times, and it was found to agree accurately with the time; the stage was then adjusted to the space corresponding to 1", upon the same supposition, and it was found to agree with the time.
Exp. 3. A third body was laid upon the horizontal plane, and a moving force applied, which descended 5½ inches in 4"; the stage was then adjusted to the space corresponding to 3", upon supposition that the spaces descended through were as the squares of the times, and it was found to agree with the time; the stage was then adjusted to the space corresponding to 2", upon the same supposition, and it was found to agree with the time; the stage was then adjusted to the space corresponding to 1", and was found to agree with the time.
Exp. 4. A fourth body was then taken and laid upon the horizontal plane, and a moving force applied, which descended 5½ inches in 4"; the stage was then adjusted to the space through which it ought to descend in 3", upon supposition that the spaces descended through were as the squares of the times, and it was found to agree with the time; the stage was then adjusted to the space corresponding to 2", upon the same supposition, and was found to agree with the time; lastly, the stage was adjusted to the space corresponding to 1", and it was found to agree exactly with the time.
Besides these experiments, a great number of others were made with hard bodies, or those whose parts so firmly cohered as not to be moved inter se by the friction; and, in each experiment, bodies of very different degrees of friction were chosen, and the results all agreed with those related above; we may therefore conclude, that the friction of hard bodies in motion is a uniformly retarding force.
But to determine whether the same was true for bodies when covered with cloth, woollen, &c., experiments were made in order to ascertain it; when it was found, in all cases, that the retarding force increased with the velocity; but, upon covering bodies with paper, the consequences were found to agree with those related above.
4. Having proved that the retarding force of all hard bodies arising from friction is uniform, the quantity of friction, considered as equivalent to a weight without inertia drawing the body on the horizontal plane backwards, or acting contrary to the moving force, may be immediately deduced from the foregoing experiments.
For let $M =$ the moving force expressed by its weight; $F =$ the friction; $W =$ the weight of the body upon the horizontal plane; $S =$ the space through which the moving force descended in the time $t$ expressed in seconds; $r = \frac{1}{6}$ feet; then the whole accelerative force (the force of gravity being unity) will be $\frac{M - F}{M + W}$. Hence, by the laws of uniformly accelerated motions,
$$\frac{M - F}{M + W} \times r^2 = S,$$
consequently $F = M - \frac{M \times W \times S}{r^2}$.
To exemplify this, let us take the case of the last experiment, where $M = 7$, $W = 25$, $S = 4$ feet, $r = \frac{1}{6}$; hence $F = 7 - \frac{32 \times 4}{16 \times 16} = 6.417$; consequently the friction was to the weight of the rubbing body as 6.4167 to 25.75. And the great accuracy of determining the friction by this method is manifest from hence, that if an error of 1 inch had been made in the descent (and experiments carefully made may always determine the space to a much greater exactness), it would not have affected the conclusion $\frac{1}{6}$th part of the whole.
5. We come in the next place to determine, whether friction, ceteris paribus, varies in proportion to the weight or pressure. Now if the whole quantity of the friction of a body, measured by a weight without inertia equivalent to the friction drawing the body backwards, increases in proportion to its weight, it is manifest, that the retardation of the velocity of the body arising from the friction will not be altered; for the retardation varies as the quantity of matter; hence, if a body be put in motion upon the horizontal plane by any moving force, if both the weight of the body and the moving force be increased in the same ratio, the acceleration arising from that moving force will remain the same, because the accelerative force varies as the moving force divided by the whole quantity of matter, and both are increased in the same ratio; and if the quantity of friction increases also as the weight, then the retardation arising from the friction will, from what has been said, remain the same, and therefore the whole acceleration of the body will not be altered; consequently the body ought, upon this supposition, still to describe the same space in the same time. Hence, by observing the spaces described in the same time, when both the body and the moving force are increased in the same ratio, we may determine whether the friction increases in proportion to the weight. The following experiments were therefore made in order to ascertain this matter:
Exp. 1. A body weighing 10 oz. by a moving force of 4 oz. described in 2" a space of 51 inches; by loading the body with 10 oz. and the moving force with 4 oz. it described 56 inches in 2"; and by loading the body again with 10 oz. and the moving force with 4 oz. it described 63 inches in 2".
Exp. 2. A body, whose weight was 16 oz. by a moving force of 5 oz. described a space of 49 inches in 3"; and by loading the body with 64 oz. and the moving force with 20 oz. the space described in the same time was 64 inches.
Exp. 3. A body weighing 6 oz. by a moving force of 2½ oz. described 28 inches in 2"; and by loading the body with 24 oz. and the moving force with 10 oz. the space described in the same time was 54 inches.
Exp. 4. A body weighing 8 oz. by a moving force of 4 oz. described 33½ inches in 2"; and by loading the body with 8 oz. and the moving force with 4 oz. the space described in the same time was 47 inches.
Exp. 5. A body whose weight was 9 oz. by a moving force of 4½ oz. described 48 inches in 2"; and by loading the body with 9 oz. and the moving force with 4½ oz. the space described in the same time was 60 inches.
Exp. 6. A body weighing 10 oz. by a moving force of 3 oz. From these experiments, and many others which it is not necessary here to relate, it appears, that the space described is always increased by increasing the weight of the body and the accelerative force in the same ratio; and as the acceleration arising from the moving force continued the same, it is manifest, that the retardation arising from the friction must have been diminished, for the whole accelerative force must have been increased on account of the increase of the space described in the same time; and hence (as the retardation from friction varies as Quantity of matter) the quantity of friction increases in a less ratio than the quantity of matter or weight of the body.
6. We come now to the last thing which it was proposed to determine, that is, whether the friction varies by varying the surface on which the body moves. Let us call two of the surfaces A and a, the former being the greater, and the latter the less. Now the weight on every given part of a is as much greater than the weight on an equal part of A, as A is greater than a; if therefore the friction was in proportion to the weight, ceteris paribus, it is manifest, that the friction on a would be equal to the friction on A, the whole friction being, upon such a supposition, as the weight on any given part of each surface multiplied into the number of such parts or into the whole area, which produces, from the proportion above, are equal. But from the last experiments it has been proved, that the friction on any given surface increases in a less ratio than the weight; consequently the friction on any given part of a has a less ratio to the friction on an equal part of A than A has to a, and hence the friction on a is less than the friction on A, that is, the smallest surface has always the least friction.
As this conclusion is contrary to the generally received opinion, Mr Vince thought it proper to confirm it by a set of experiments made with different bodies of exactly the same degree of roughness on their two surfaces.
Exp. 1. A body was taken whose flat surface was to its edge as 22:9, and with the same moving force the body described on its flat side 33\(\frac{1}{4}\) inches in 2\(\frac{1}{2}\)s, and on its edge 47 inches in the same time.
Exp. 2. A second body was taken whose flat surface was to its edge as 32:3, and with the same moving force it described on its flat side 32 inches in 2\(\frac{1}{2}\)s, and on its edge it described 57\(\frac{1}{4}\) inches in the same time.
Exp. 3. He took another body and covered one of its surfaces, whose length was 9 inches, with a fine rough paper, and by applying a moving force, it described 25 inches in 2\(\frac{1}{2}\)s; he then took off some paper from the middle, leaving only \(\frac{1}{2}\) of an inch at the two ends, and with the same moving force it described 40 inches in the same time.
Exp. 4. Another body was taken which had one of its surfaces, whose length was 9 inches, covered with a fine rough paper, and by applying a moving force it described 42 inches in 2\(\frac{1}{2}\)s; some of the paper was then taken off from the middle, leaving only 1\(\frac{1}{2}\) inches at the two ends, and with the same moving force it described 54 inches in 2\(\frac{1}{2}\)s; he then took off more paper, leaving only \(\frac{1}{2}\) of an inch at the two ends, and the body then described, by the same moving force, 60 inches in the same time.
In the two last experiments the paper which was taken off the surface was laid on the body, that its weight might not be altered.
Exp. 5. A body was taken whose flat surface was to its edge as 30:17; the flat side was laid upon the horizontal plane, a moving force was applied, and the stage was fixed in order to stop the moving force, in consequence of which the body would then go on with the velocity acquired until the friction had destroyed all its motion; when it appeared from a mean of 12 trials that the body moved, after its acceleration ceased, 5\(\frac{1}{2}\) inches before it stopped. The edge was then applied, and the moving force descended through the same space; and it was found, from a mean of the same number of trials, that the space described was 7\(\frac{1}{2}\) inches before the body lost all its motion, after it ceased to be accelerated.
Exp. 6. Another body was then taken whose flat surface was to its edge as 60:19, and, by proceeding as before, on the flat surface it described, at a mean of 12 trials, 5\(\frac{1}{2}\) inches, and on the edge 6\(\frac{1}{2}\) inches, before it stopped, after the acceleration ceased.
Exp. 7. Another body was taken whose flat surface was to its edge as 26:3, and the spaces described on the two surfaces, after the acceleration ended, were, at a mean of ten trials, 4\(\frac{1}{2}\) and 7\(\frac{1}{2}\) inches respectively.
From all these different experiments it appears, that the smallest surface had always the least friction, which agrees with the consequence deduced from the consideration that the friction does not increase in so great a ratio as the weight; we may therefore conclude, that the friction of a body does not continue the same when it has different surfaces applied to the plane on which it moves, but that the smallest surface will have the least friction.
To the experiments instituted by Mr Ferguson and others, from which conclusions have been drawn to different from these, our author makes the following objections: It was their object to find what moving force would just put a body at rest in motion; and having, as they thought, found it, they thence concluded, that the accelerative force was then equal to the friction. But it is manifest, as Mr Vince observes, that any force which will put a body in motion must be greater than the force which opposes its motion, otherwise it could not overcome it; and hence, if there were no other objection than this, it is evident, that the friction could not be very accurately obtained: but there is another objection which totally destroys the experiment so far as it tends to shew the quantity of friction, which is the strong cohesion of the body to the plane when it rests at rest; and this is confirmed by the following experiments. If a body of 12\(\frac{1}{2}\) oz. was laid upon an horizontal plane, and then loaded with a weight of 8lb. and such a moving force was applied as would, when the body was just put in motion, continue that motion without any acceleration; in which case the friction must be just equal to the accelerative force. The body was then stopped, when it appeared, that the same moving force which had kept the body in motion before, fore, would not put it in motion, and it was found necessary to take off 4½ oz. from the body before the same moving force would put it in motion; it appears therefore, that this body, when laid upon the plane, at rest, acquired a very strong cohesion to it.
A body whose weight was 16 oz. was laid at rest upon the horizontal plane, and it was found that a moving force of 6 oz. would just put it in motion; but that a moving force of 4 oz. would, when it was just put in motion, continue that motion without any acceleration, and therefore the accelerative force must then have been equal to the friction, and not when the moving force of 6 oz. was applied.
From these experiments therefore it appears, how very considerable the cohesion was in proportion to the friction when the body was in motion; it being, in the latter case, almost ¼d., and in the former it was found to be very nearly equal to the whole friction. All the conclusions therefore deduced from the experiments, which have been instituted to determine the friction from the force necessary to put a body in motion, have manifestly been totally false; as such experiments only shew the reliance which arises from the cohesion and friction conjointly.
Our author concludes this part of his subject with the following remark upon n° 5: "It appears from all the experiments (says he) which I have made, that the proportion of the increase of the friction to the increase of the weight was different in all the different bodies which were made use of; no general rule therefore can be established to determine this for all bodies, and the experiments which I have hitherto made have not been sufficient to determine it for the same body."
He then proceeds to establish a theory upon the principles which he has deduced from his experiments. That theory is comprehended in five propositions, of which the object of the first is "to find the time of descent, and the number of revolutions made by a cylinder rolling down an inclined plane in consequence of its friction.
II. "To determine the space through which a body, projected on an horizontal plane with a given velocity, will move before it stops, or before its motion becomes uniform.
III. "To find the centre of friction.
IV. "To determine, from the given velocity with which a body begins to revolve about the centre of its base, the number of revolutions which that body will make before all its motion be destroyed.
V. "To find the nature of the curve described by any point of a body affected by friction when it descends down any inclined plane."
To give the solutions of these problems, with the corollaries deduced from them, would swell this article to very little purpose; for they would be unintelligible to the mere mechanic, and the mathematician will either solve them for himself, or have recourse to the original memoir, where he will find solutions at once elegant and perspicuous.
FRIGORIFIC Mixtures, are those which experience has taught philosophers to employ for the purpose of producing artificial cold. Some of these mixtures are enumerated under the title COLD (Encycl.), and a much more accurate list of them is given, together with the principle upon which they produce their effect, in the article CHEMISTRY, n° 282. (Suppl.)
There is one mixture, however, not mentioned in that list, which was employed by Seguin, and seems, on many accounts, to be the most eligible that has yet been proposed. Considering the muriatic (see Chemistry-Index, Suppl.) as a class of salts best suited for the purpose, he gave the decided preference to muriatic of lime in crystals; and his method was to mix the crystals, previously pulverised, with an equal weight of uncompressed snow.
By means of this mixture Mr W. H. Pepys junior, of the London Philosophical Society, with the affiance of some friends, froze, on the 8th of February 1799, 56 lbs. avoirdupoise of mercury into a solid mass. The mercury was put into a strong bladder and well secured at the mouth, the temperature of the laboratory at the time being +33°. A mixture consisting of muriatic of lime 2 lb. at +33°, and the same weight of snow at +32° gave —42° (a). The mercury was put as gently as possible into this mixture (to prevent a rupture of the bladder); by means of a cloth held at the four corners. When the cold mixture had robbed the mercury of so much of its heat as to have its own temperature thereby raised from —42° to +5°, another mixture, the same in every respect as the last, was made, which gave, on trial with the thermometer, —43°. The mercury was now received into the cloth, and put gently into this new mixture, where it was left to be cooled still lower than before.
In the mean time five pounds of muriatic of lime, in a large pail made of tinned iron, and japanned inside and outside, was placed in a cooling mixture in an earthenware pan. The mixture in the pan, which consisted of 4 lb. of muriatic of lime and a like quantity of snow, of the same temperature as the former, in one hour reduced the 5 lb. of muriatic in the pail to —15°. The mixture was then emptied out of the earthen pan, and four large corks, at proper distances, placed on its bottom, to serve as rests for the japanned pail which was now put into the pan. The corks answered the purpose of insulating the inner vessel, while the exterior one kept off the surrounding atmosphere, and preserved the air between the two at a low temperature.
To the 5 lb. of muriatic of lime which had been cooled, as already noticed, to —15°, and which still remained in the metallic vessel, was now added snow, uncompressed and free from moisture, at the usual temperature of +32°. In less than three minutes the mixture gave a temperature of —62°: a degree of cold which perhaps was never before produced in this country, being 94° below the freezing point of water.
The mercury, which, by immersion in the second cooling mixture to which it was exposed, was, by this time reduced to —30°, was now, by the means employed before, cautiously put into the last made mixture of the temperature of —62°. A hoop, with net-work fastened to its upper edge, and of such a breadth in the rim... rim that the net-work, when loaded with the bladder of mercury, could not reach its lower edge, was at the bottom of the mixture, to prevent the bladder from coming in contact with the vessel; by which means the mercury was suspended in the middle of the mixture.
As soon as the bladder was safely deposited on the net-work, the vessels were carefully covered over with a cloth, to impede the passage of heat from the surrounding atmosphere into the freezing materials. The condensation of moisture from the atmosphere by the agency of so low a temperature was greater than could have been expected: it floated like steam over the vessels, and, but for the interposed covering, would have given the mixture more temperature than was desirable.
After one hour and forty minutes they found, by means of a search introduced for the purpose, that the mercury was solid and fixed. The temperature of the mixture at this time was —46°, that is, 16° higher than when the mercury was put into it.
Our young philosophers having neglected to sling the hoop and net-work in such a manner as might have enabled them to lift it out of the mixture at once, with the bladder and its contents, were obliged to turn out the whole contents of the pail into a large evaporating capsule made of iron. This was not effected without the mercury striking against its bottom and being fractured, though it received a considerable increase of temperature from the capsule. The fracture was similar to that of zinc, but with parts more cubical. The larger pieces were kept for some minutes before fusion took place, while others were twisted and bent into various forms, to the no small gratification and surprise of those who had never witnessed or expected to see such an effect produced on so fusible a metal.
In experiments of the kind here described, all the exterior vessels should be of earthen-ware or wood, which being bad conductors of heat, prevent the ingredients from receiving heat from the atmosphere and surrounding objects with the same facility that they would through metals; and, for a similar reason, the interior vessels are best of metal, that they may allow the heat to pass more readily from the substance to be cooled into the frigorific mixture employed for that purpose.
Muriat of lime is certainly the most powerful, and at the same time the most economical substance that can be employed for producing artificial cold; for its first cost is a mere trifle, being a residuum from many chemical processes, as the distillation of pure ammonia, &c., and often thrown away: besides, it may be repeatedly used for similar experiments, nothing being necessary for this purpose but filtration and evaporation to bring it to its first state. The evaporation should be carried on till the solution becomes as thick as a strong syrup, and upon cooling the whole will be crystallized; it must then be powdered, put up in dry bottles, well corked, and covered with bladder or cement to prevent liquefaction; which otherwise would soon take place, owing to the great affinity the muriat has for moisture.
The powerful effects produced by the frigorific mixture of muriat of lime and snow, present a wide field for experiments to determine the possibility of fixing some of the gates by intense cold. And we are happy to be informed by Mr Pepys, that, as soon as an opportunity offers, he and his friends mean to make some experiments with that view, and to communicate the result of them to the editor of the valuable miscellany * from which we have taken this account of his experiment on mercury.