| Number of Experiment. | Total load on the slide in lbs. | Angle of elevation. 1st Series. | Angle of elevation. 2nd Series. | Mean values of the angles. |
|---|---|---|---|---|
| 1 | 14 | 19°·5 | —— | 19°·5 |
| 2 | 56 | 20°·1 | 17°·2 | 18°·6 |
| 3 | 112 | 20°·3 | 18°·9 | 19°·6 |
145. We might, however, proceed differently in determining the angle of friction, by giving the load a start, and ascertaining if the motion will continue. To do so requires the aid of an assistant, who will start the load with the help of the screw, while the elevation of the plane is being slowly increased. The result of these experiments is given in Table VIII.
Table VIII.—Angle of Friction.
A smooth plane of pine 72" × 11" carries a loaded slide of pine 9" × 9"; one end of the plane is gradually elevated until the slide, having received a start, moves off uniformly.
| Number of Experiment. | Total load on slide in lbs. | Angle of inclination. |
|---|---|---|
| 1 | 14 | 14°·3 |
| 2 | 56 | 13°·0 |
| 3 | 112 | 13°·0 |
We see from this table also that the angle of friction is independent of the load, but the angle is in this case less by 5° or 6° than was found necessary to impart motion when a start was not given.
146. It is commonly stated that the coefficient of friction equals the tangent of the angle of friction, and this can be proved to be true when the ordinary law of friction is assumed. But as we have seen that the law of friction is only approximately correct, we need not expect to find this other law completely verified.
147. When the slide is started, the mean value of the angle of friction is 13°·4. The tangent of this angle is 0·24: this is about 11 per cent. less than the coefficient of friction 0·27, which we have already determined. The mean value of the angle of friction when the slide is not started is 19°·2, and its tangent is 0·35. The experiments of [Table I]. are, as already pointed out, rather unsatisfactory, but we refer to them here to show that, so far as they go, the coefficient of friction is in no sense equal to the tangent of the angle of friction. If we adopt the mean values given in the last column of [Table I]., the best coefficient of friction which can be deduced is 0·41. Whether, therefore, the slide be started or not started, the tangent of the angle of friction is smaller than the corresponding coefficient of friction. When the slide is started, the tangent is about 11 per cent. less than the coefficient; and when the slide is not started, it is about 14 per cent. less. There are doubtless many cases in which these differences are sufficiently small to be neglected, and in which, therefore, the law may be received as true.
ANOTHER LAW OF FRICTION.
148. The area of the wooden slide is 9" × 9", but we would have found that the friction under a given load was practically the same whatever were the area of the slide, so long as its material remained unaltered. This follows as a consequence of the approximate law that the friction is proportional to the pressure. Suppose that the weight were 100 lbs., and the area of the slide 100 inches, there would then be a pressure of 1 lb. per square inch over the surface of the slide, and therefore the friction to be overcome on each square inch would be 0·27 lb., or for the whole slide 27 lbs. If, however, the slide had only an area of 50 square inches, the load would produce a pressure of 2 lbs., per square inch; the friction would therefore be 2 × 0·27 = 0·54 lb. for each square inch, and the total friction would be 50 × 0·54 = 27 lbs., the same as before: hence the total friction is independent of the extent of surface. This would remain equally true even though the weight were not, as we have supposed, uniformly distributed over the surface of the slide.