136. We are therefore led to inquire whether some other relation between F and R may not represent the experiments with greater fidelity than the common law of friction. If we diminished the coefficient by a small amount, and then added a constant quantity to the product of the coefficient and the load, the effect of this change would be that for small loads the calculated values would be increased, while for large loads they would be diminished. This is the kind of change which we have indicated to be necessary for reconciliation between the observed and calculated values.
137. We therefore infer that a relation of the form F = x + y R will probably express a more correct law, provided we can find x and y. One equation between x and y is obtained by introducing any value of R with the corresponding value of F, and a second equation can be found by taking any other similar pair. From these two equations the values of x and of y may be deduced by elementary algebra, but the best formula will be obtained by combining together all the pairs of corresponding values. For this reason the method described in the Appendix must be used, which, as it is founded on all the experiments, must give a thoroughly representative result. The formula thus determined, is
F = 1·44 + 0·252 R.
This formula is compared with the experiments in Table IV.
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 = 1·44 + 0·252 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 | 5·0 | +0·3 |
| 2 | 28 | 8·2 | 8·5 | +0·3 |
| 3 | 42 | 12·2 | 12·0 | -0·2 |
| 4 | 56 | 15·8 | 15·6 | -0·2 |
| 5 | 70 | 19·4 | 19·1 | -0·3 |
| 6 | 84 | 23·0 | 22·6 | -0·4 |
| 7 | 98 | 25·8 | 26·1 | +0·3 |
| 8 | 112 | 29·3 | 29·7 | +0·4 |
The fourth column contains the calculated values: thus, for example, in experiment 4, where the load is 56 lbs., the calculated value is 1·44 + 0·252 × 56 = 15·6; the difference 0·2 between this and the observed value 15·8 is shown in the last column.
138. It will be noticed that the greatest discrepancy in this column is 0·4 lbs., and that therefore the formula represents the experiments with considerable accuracy. It is undoubtedly nearer the truth than the former law ([Art. 132]); in fact, the differences are now such as might really belong to errors unavoidable in making the experiments.
139. This formula may be used for calculating the friction for any load between 14 lbs. and 112 lbs. Thus, if the load be 63 lbs., the friction is 1·44 + 0·252 × 63 = 17·3 lbs., which does not differ much from 17·0 lbs., the value found by the more ordinary law. We must, however, be cautious not to apply this formula to weights which do not lie between the limits of the greatest and least weight used in those experiments by which the numerical values in the formula have been determined; for example, to take an extreme case, if R = 0, the formula would indicate that the friction was 1·44, which is evidently absurd; here the formula errs in excess, while if the load were very large it is certain the formula would err in defect.