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.
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.
THE COEFFICIENT VARIES WITH
THE WEIGHTS USED.
140. In a subsequent lecture we shall employ as an inclined plane the plank we have been examining, and we shall require to use the knowledge of its friction which we are now acquiring. The weights which we shall then employ range from 7 lbs. to 56 lbs. Assuming the ordinary law of friction, we have found that 0·27 is the best value of its coefficient when the loads range between 14 lbs. and 112 lbs. Suppose we only consider loads up to 56 lbs., we find that the coefficient 0·288 will best represent the experiments within this range, though for 112 lbs. it would give an error of nearly 3 lbs. The results calculated by the formula F = 0·288 R are shown in Table V., where the greatest difference is 0·7 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 = 0·288 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 | 4·0 | -0·7 |
| 2 | 28 | 8·2 | 8·1 | -0·1 |
| 3 | 42 | 12·2 | 12·1 | -0·1 |
| 4 | 56 | 15·8 | 16·1 | +0·3 |
141. But we can replace the common law of friction by the more accurate law of [Art. 137], and the formula computed so as to best harmonise the experiments up to 56 lbs., disregarding the higher loads, is F = 0·9 + 0·266 R. This formula is obtained by the method referred to in [Art. 137]. We find that it represents the experiments better than that used in [Table V]. Between the limits named, this formula is also more accurate than that of [Table IV]. It is compared with the experiments in [Table VI]., and it will be noticed that it represents them with great precision, as no discrepancy exceeds 0·1.
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 = 0·9 + 0·266 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 | 4·6 | -0·1 |
| 2 | 28 | 8·2 | 8·3 | +0·1 |
| 3 | 42 | 12·2 | 12·1 | -0·1 |
| 4 | 56 | 15·8 | 15·8 | 0·0 |
Fig. 33.