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).