250. The lever of the third order may be easily understood from Fig.
39, of which we have already made use. In the lever of the third
order the fulcrum is at one end, the load is at the other end, while
the power lies between the two. In this case, then, the power is
represented by the 56 lb. weight, while the load is indicated by the
spring balance. The power always exceeds the load, and consequently
this lever is to be used where speed is to be gained instead of power.
Thus, for example; when the power, 56 lbs., is 2' distant from the
fulcrum, the load indicated by the spring balance is about 23 lbs.
251. The treadle of a grindstone is often a lever of the third order.
The fulcrum is at one end, the load is at the other end, and the foot
has only to move through a small distance.
252. The principles which have been discussed in Lecture III. with
respect to parallel forces explain the laws now laid down for levers of
different orders, and will also enable us to express these laws more
concisely.
253. A comparison between Figs. 20 and 39 shows that the only
difference between the contrivances is that in Fig. 20 we have a
spring balance C in the same place as the steel edge A in Fig 39. We
may in Fig. 20 regard one spring balance as the power, the other as
the fulcrum, and the weight as the load. Nor is there any essential
difference between the apparatus of Fig. 38 and that of Fig. 20. In
Fig. 38 the bar is pulled down by a force at each end, one a weight,
the other a spring balance, while it is supported by the upward
pressure of the steel edge. In Fig. 20 the bar is being pulled upwards
by a force at each end, and downwards by the weight. The two cases are
substantially the same. In each of them we find a bar acted upon by a
pair of parallel forces applied at its extremities, and retained in
equilibrium by a third force.
254. We may therefore apply to the lever the principles of parallel
forces already explained. We showed that two parallel forces acting
upon a bar could be compounded into a resultant, applied at a certain
point of the bar. We have defined the moment of a force (Art. 64),
and proved that the moments of two parallel forces about the point of
application of their resultant are equal.
255. In the lever of the first order there are two parallel forces, one
at each end; these are compounded into a resultant, and it is necessary
that this resultant be applied exactly over the steel edge or fulcrum
in order that the bar may be maintained at rest. In the levers of the
second and third orders, the power and the load are two parallel forces
acting in opposite directions; their resultant, therefore, does not
lie between the forces, but is applied on the side of the greater, and
at the point where the steel edge supports the bar. In all cases the
moment of one of the forces about the fulcrum must be equal to that of
the other. From the equality of moments it follows that the product of
the power and the distance of the power from the fulcrum equals the
product of the load, and the distance of the load from the fulcrum:
this principle suffices to demonstrate the rules already given.
256. The laws governing the lever may be deduced from the principle
of work; the load, if nearer than the power to the fulcrum, is moved
through a smaller distance than the power. Thus, for example, in
the lever of the first order: if the load be 12 times as far as the
power from the fulcrum, then for every inch the load moves it can be
demonstrated that the power must move 12 inches. The number of units
of work applied at one end of a machine is equal to the number yielded
at the other, always excepting the loss due to friction, which is,
however, so small in the lever that we may neglect it. If then a power
of 1 lb. be applied to move the power end through 12 inches, one unit
of work will have been put into the machine. Hence one unit of work
must be done on the load, but the load only moves through ¹/₁₂ of a
foot, and therefore a load of 12 lbs. could be overcome: this is the
same result as would be given by the rule (Art. 236).
257. To conclude: we have first determined by actual experiment the
relation between the power and the load in the lever; we have seen that
the law thus obtained harmonizes with the principle of the composition
of parallel forces; and, finally, we have shown how the same result can
be deduced from the fertile and important principle of work.
LECTURE IX.
_THE INCLINED PLANE AND THE SCREW._
The Inclined Plane without Friction.—The
Inclined Plane with Friction.—The Screw.—The
Screw-jack.—The Bolt and Nut.