THE WHEEL AND PINION.

326. By means of what are called cog-wheels or toothed wheels, we are enabled to combine two or more wheels and axles together, and thus greatly to increase the power which can be produced by a single wheel and axle. Toothed wheels are used for a great variety of purposes in mechanics; we have already had some illustration of their use during these lectures ([Fig. 30]). The wheels which we shall employ are those often used in lathes and other small machines; they are what are called 10-pitch wheels,—that is to say, a wheel of this class contains ten times as many teeth in its circumference as there are inches in its diameter. I have here a wheel 20" diameter, and consequently it has 200 teeth; here is another which is 2"·5 diameter, and which consequently contains 25 teeth. We shall mount these wheels upon two parallel shafts, so that they gear one into the other in the manner shown in [Fig. 46]: f is the large wheel containing 200 teeth, and g the pinion of 25 teeth. The axles are 0"·75 diameter; around each of them a rope is wound, by which a hook is suspended.

327. A small weight at k is sufficient to raise a much larger weight on the other shaft; but before experimenting on the mechanical efficiency of this arrangement, we shall as usual calculate the velocity ratio. The wheel contains eight times as many teeth as the pinion; it is therefore evident that when the wheel has made one revolution, the pinion will have made eight revolutions, and conversely the pinion must turn round eight times to turn the wheel round once: hence the power which is turning the pinion round must be lowered through eight times the circumference of the axle, while the load is raised through a length equal to one circumference of the axle. We thus find the velocity ratio of the machine to be 8.

328. We determine the mechanical efficiency by trial. Attaching a load of 56 lbs. to the axle of the large wheel, it is observed that a power of 13·7 lbs. at k will raise it; the mechanical efficiency of the machine is therefore about 4·1, which is almost exactly half the velocity ratio. We note that the load will only just run down when the power is removed; from this we might have inferred, by [Art. 222], that nearly half the power is expended on friction, and that therefore the mechanical efficiency is about half the velocity ratio. The actual percentage of energy that is utilised with this particular load is 51. If we suspend 112 lbs. from the load hook, 26 lbs. is just enough to raise it; the mechanical efficiency that would be deduced from this result is 112 ÷ 26 = 4·3, which is slightly in excess of the amount obtained by the former experiment. It is often found to be a property of the mechanical powers, that as the load increases the mechanical efficiency slightly improves.

329. In [Table XX]. will be found a record of experiments upon the relation between the power and the load with the wheel and pinion; the table will sufficiently explain itself, after the description of similar tables already given ([Arts. 310], [324]).

Table XX.—The Wheel and Pinion.

Wheel (10-pitch), 200 teeth; pinion, 25 teeth; axles equal, effective circumference of each being 2"·87; oiled brass bearings; velocity ratio, 8; mechanical efficiency, 4·1; useful effect, 51 per cent.; formula, P = 2·46 + 0·21 R.

Number of
Experiment. 		R.
Load
in lbs. 		Observed
power
in lbs.		P.
Calculated
power
in lbs.		Difference of the
observed and
calculated powers.
114 5·4 5·4 0·0
228 8·7 8·3-0·4
34211·011·3+0·3
45613·714·2+0·5
57017·517·2-0·3
68420·020·1+0·1
79823·023·0 0·0
8112 26·026·0 0·0

330. The large amount of friction present in this contrivance is the consequence of winding the rope directly upon the axle instead of upon a barrel, as already pointed out in [Art. 319]. We might place barrels upon these axles and demonstrate the truth of this statement; but we need not delay to do so, as we use the barrel in the machines which we shall next describe.