335. A series of experiments made with this crane is recorded in [Table XXI]., and a comparison of the calculated and observed values will show that the formula P = 0·0556 R represents the experiments with considerable accuracy.
336. It may be noticed that in this formula the term independent of R, which we frequently meet with in the expression of the relation between the power and the load, is absent. The probable explanation is to be found in the fact that some minute irregularity in the form of the barrel or of the wheel has been constantly acting like a small weight in favour of the power. In each experiment the motion is always started from the same position of the wheels, and hence any irregularity will be constantly acting in favour of the power or against it; here the former appears to have happened. In other cases doubtless the latter has occurred; the difference is, however, of extremely small amount. The friction of the machine itself when without a load is another source for the production of the constant term; it has happened in the present case that this friction has been almost exactly balanced by the accidental influence referred to.
337. In cranes it is usual to provide means of adding a second train of wheels, when the load is very heavy. In another model we applied the power to an axle with a pinion of 25 teeth, gearing into a wheel of 200 teeth; on the axle of the wheel with 200 teeth is a pinion of 30 teeth, which gears into a wheel of 180 teeth; the barrel is on the axle of the last wheel. A series of experiments with this crane is shown in Table XXII.
Table XXII.—The Crane for Heavy Loads.
Circumference of wheel to which power is applied, 43"; train of wheels, 25 ÷ 200 × 30 ÷ 180; circumference of drum on which rope is wound, 14"·9; velocity ratio, 137; mechanical efficiency, 87; useful effect, 63 per cent.; formula, P = 0·185 + 0·00782 R.
| Number of Experiment. | R. Load in lbs. | Observed power in lbs. | P. Calculated power in lbs. | Difference of the observed and calculated powers. |
|---|---|---|---|---|
| 1 | 14 | 0·30 | 0·29 | -0·01 |
| 2 | 28 | 0·40 | 0·40 | 0·00 |
| 3 | 42 | 0·50 | 0·51 | +0·01 |
| 4 | 56 | 0·60 | 0·62 | +0·02 |
| 5 | 70 | 0·75 | 0·73 | -0·02 |
| 6 | 84 | 0·85 | 0·84 | -0·01 |
| 7 | 98 | 0·95 | 0·95 | 0·00 |
| 8 | 112 | 1·05 | 1·06 | +0·01 |
The velocity ratio is now 137, and the mechanical efficiency is 87; one man could therefore raise a ton with ease by applying a power of 26 lbs. to a crane of this kind.
CONCLUSION.
338. It will be useful to contrast the wheel and axle on which we have experimented ([Art. 304]) with the differential pulley ([Art. 209]). The velocity ratio of the former machine is nearly double that of the latter, and its mechanical efficiency is nearly four times as great. Less than half the applied power is wasted in the wheel and axle, while more than half is wasted in the differential pulley. This makes the wheel and axle both a more powerful machine, and a more economical machine than the differential pulley. On the other hand, the greater compactness of the latter, its facility of application, and the practical conveniences arising from the property of not allowing the load to run down, do often more than compensate for the superior mechanical advantage of the wheel and axle.
339. We may also contrast the wheel and axle with the screw ([Art. 277]). The screw is remarkable among the mechanical powers for its very high velocity ratio, and its excessive friction. Thus we have seen in [Art. 291] how the velocity ratio of a screw-jack with an arm attached amounted to 414, while its mechanical efficiency was little more than one-fourth as great. No single wheel and axle could conveniently be made to give a mechanical efficiency of 116; but from [Art. 337] we could easily design a combination of wheels and axles to yield an efficiency of quite this amount. The friction in the wheel and axle is very much less than in the screw, and consequently energy is saved by the use of the former machine.