131. To test this supposition we must try to determine k; it may be ascertained by dividing any value of F by the corresponding value of R. If this be done, we shall find that each of the experiments yields a different quotient; the first gives 0·336, and the last 0·262, while the other experiments give results between these extreme values. These numbers are tolerably close together, but there is still sufficient discrepancy to show that it is not strictly true to assert that the friction is proportional to the load.
132. But the law that the friction varies proportionally to the pressure is so approximately true as to be sufficient for most practical purposes, and the question then arises, which of the different values of k shall we adopt? By a method which is described in the Appendix we can determine a value for k which, while it does not represent any one of the experiments precisely, yet represents them collectively better than it is possible for any other value to do. The number thus found is 0·27. It is intermediate between the two values already stated to be extreme. The character of this result is determined by an inspection of [Table III].
The fourth column of this table has been calculated from the formula F = 0·27 R. Thus, for example, in experiment 5, the friction of a load of 70 lbs. is 19·4 lbs., and the product of 70 and 0·27 is 18·9, which is 0·5 lb. less than the true amount. In the last column of this table the discrepancies between the observed and the calculated values are recorded, for facility of comparison. It will be observed that the greatest difference is under 1 lb.
Friction of pine upon pine; the mean values of the friction given in [Table II]. (corrected for the friction of the pulley) compared with the formula F = 0·27 R.
| Number of Experiment. | R. Total load on slide in lbs. | Corrected mean value of friction. | F. Calculated value of friction. | Discrepancies between the observed and calculated frictions. |
|---|---|---|---|---|
| 1 | 14 | 4·7 | 3·8 | -0·9 |
| 2 | 28 | 8·2 | 7·6 | -0·6 |
| 3 | 42 | 12·2 | 11·3 | -0·9 |
| 4 | 56 | 15·8 | 15·1 | -0·7 |
| 5 | 70 | 19·4 | 18·9 | -0·5 |
| 6 | 84 | 23·0 | 22·7 | -0·3 |
| 7 | 98 | 25·8 | 26·5 | +0·7 |
| 8 | 112 | 29·3 | 30·2 | +0·9 |
133. Hence the law F = 0·27 R represents the experiments with tolerable accuracy; and the numerical ratio O·27 is called the coefficient of friction. We may apply this law to ascertain the friction in any case where the load lies between 14 lbs. and 112 lbs.; for example, if the load be 63 lbs., the friction is 63 × 0·27 = 17·0.
134. The coefficient of friction would have been slightly different had the grain of the slide been parallel to that of the plank; and it of course varies with the nature of the surfaces. Experimenters have given tables of the coefficients of friction of various substances, wood, stone, metals, &c. The use of these coefficients depends upon the assumption of the ordinary law of friction, namely, that the friction is proportional to the pressure: this law is accurate enough for most purposes, especially when used for loads that lie between the extreme weights employed in calculating the value of the coefficient which is employed.
A MORE ACCURATE LAW OF FRICTION.
135. In making one of our measurements with care, it is unusual to have an error of more than a few tenths of 1 lb. and it is hardly possible that any of the mean frictions we have found should be in error to so great an extent as 0·5 lb. But with the value of the coefficient of friction which is used in [Table III]., the discrepancies amount sometimes to 0·9 lbs. With any other numerical coefficient than 0·27, the discrepancies would have been even still more serious. As these are too great to be attributed to errors of experiment, we have to infer that the law of friction which has been assumed cannot be strictly true. The signs of the discrepancies indicate that the law gives frictions which for small loads are too small, and for large loads are too great.