NICS Sect

NICS 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. 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 stage, 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. Every thing 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 $5\backslash frac\{1\}\{2\}$ inches in 4"; for by the beat of the clock, and the sound of the moving force when it arrived at the stage, the space could be very accurately adjusted to the time: The stage 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 stage 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 stage 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 stage 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

Friction. 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 $41\backslash frac\{1\}\{2\}$ 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 59$\backslash frac\{1\}\{2\}$ 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 55 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; $r$ = 16$\backslash frac\{1\}\{2\}$ feet; then the whole accelerative force (the force of gravity being unity) will be $\backslash frac\{M-F\}\{M+W\}$; hence, by the laws of uniformly accelerated motions, $\backslash frac\{M-F\}\{M+W\}\; \backslash times\; rt^2\; =\; S$, consequently $F\; =\; M\; -\; \backslash frac\{M\; \backslash times\; W\; \backslash times\; S\}\{rt^2\}$.

To exemplify this, let us take the case of the last ex-

periment, where $M\; =\; 7$, $W\; =\; 25\backslash frac\{1\}\{2\}$, $S\; =\; 4\backslash frac\{1\}\{2\}$ feet, $rt^2\; =\; 4\text{'}\text{'}$; hence $F\; =\; 7\; -\; \backslash frac\{32\backslash frac\{1\}\{2\}\; \backslash times\; 4\backslash frac\{1\}\{2\}\}\{16\backslash frac\{1\}\{2\}\; \backslash times\; 16\}\; =\; 6.417$; 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 $\backslash frac\{1\}\{350\}$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 $\backslash frac\{\backslash text\{Quantity\; of\; friction\}\}\{\backslash text\{Quantity\; of\; matter\}\}$; 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 2$\backslash frac\{1\}\{2\}$ 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 33$\backslash frac\{1\}\{2\}$ 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 4$\backslash frac\{1\}\{2\}$ oz. described 48 inches in 2"; and by loading the body with 9 oz. and the moving force with 4$\backslash frac\{1\}\{2\}$ 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.

3 oz. described 20 inches in 2"; by loading the body with 10 oz. and the moving force with 3 oz. the space described in the same time was 31 inches; and by loading the body again with 30 oz. and the moving force with 9 oz. the space described was 34 inches in 2".

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 $\backslash frac\{\backslash text\{Quantity\; of\; friction\}\}\{\backslash text\{Quantity\; of\; matter\}\}$) 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$\backslash frac\{1\}\{2\}$ inches in 2", 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", and on its edge it described 37$\backslash frac\{1\}\{2\}$ 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"; he then took off some paper from the middle, leaving only $\backslash frac\{1\}\{2\}$ 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"; some of the paper was then taken off from the middle, leaving only $\backslash frac\{1\}\{2\}$ inches at

the two ends, and with the same moving force it described 54 inches in 2"; he then took off more paper, leaving only $\backslash frac\{1\}\{4\}$ 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$\backslash frac\{1\}\{2\}$ 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$\backslash frac\{1\}\{2\}$ 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$\backslash frac\{1\}\{2\}$ inches, and on the edge 6$\backslash frac\{1\}\{2\}$ 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 these two surfaces, after the acceleration ended, were, at a mean of ten trials, 4$\backslash frac\{1\}\{2\}$ and 7$\backslash frac\{1\}\{2\}$ 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 so 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 lies at rest; and this is confirmed by the following experiments. 1$\backslash frac\{1\}\{2\}$. A body of 12$\backslash frac\{1\}\{2\}$ oz. was laid upon an horizontal plane, and then loaded with a weight of 8 lb. 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 be-

Friction, fore, would not put it in motion, and it was found necessary to take off $4\backslash frac\{1\}\{2\}$ 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. 2dly, 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 $\backslash frac\{1\}\{2\}$d, 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 resistance which arises from the cohesion and friction conjointly.

Our author concludes this part of his subject with the following remark upon n^o 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.