161. There is often a considerable advantage obtained by using large
rather than small pulleys. The amount of force necessary to overcome
friction varies inversely as the size of the pulley. We shall
demonstrate this by actual experiment with the apparatus of Fig. 34. A
small pulley K is attached to the large pulley I; they are in fact one
piece, and turn together on the same axle. Hence if we first determine
the friction with the rope over the large pulley, and then with the
rope over the small pulley, any difference can only be due to the
difference in size, as all the other circumstances are the same.
162. In making the experiments we must attend to the following point.
The pulleys and the socket on which they are mounted weigh several
pounds, and consequently there is friction on the axle arising from
the weight of the pulleys, quite independently of any weights that may
be placed on the hooks. We must then, if possible, evade the friction
of the pulley itself, so that the amount of friction which is observed
will be entirely due to the weights raised. This can be easily done.
The rope and hooks being on the large pulley I, I find that 0·16 lb.
attached to one of the hooks, E, is sufficient to overcome the friction
of the pulley, and to make that hook descend and raise F. If therefore
we leave 0·16 lb. on E, we may consider the friction due to the weight
of the pulley, rope, and hooks as neutralized.
[Illustration: FIG. 34.]
163. I now place a stone weight on each of the hooks E and F. The
amount necessary to make the hook E and its load descend is 0·28 lb.
This does not of course include the weight of 0·16 lb. already referred
to. We see therefore that with the large pulley the amount of friction
to be overcome in raising one stone is 0·28 lb.
164. Let us now perform precisely the same experiment with the small
pulley. I transfer the same rope and hooks to K, and I find that 0·16
lb. is not now sufficient to overcome the friction of the pulley, but
I add on weights until C will just descend, which occurs when the load
reaches 0·95 lb. This weight is to be left on C as a counterpoise,
for the reasons already pointed out. I place a stone weight on C and
another on D, and you see that C will descend when it receives an
additional load of 1·35 lbs.; this is therefore the amount of friction
to be overcome when a stone weight is raised over the pulley K.
165. Let us compare these results with the dimensions of the pulleys.
The proper way to measure the effective circumference of a pulley when
carrying a certain rope is to measure the length of that rope which
will just embrace it. The length measured in this way will of course
depend to a certain extent upon the size of the rope. I find that the
circumferences of the two pulleys are 43"·0 and 9"·5. The ratio of
these is 4·5; the corresponding resistances from friction we have seen
to be 0·28 lb. and 1·35 lbs. The larger of these quantities is 4·8
times the smaller. This number is very close to 4·5; we must not, as
already explained, expect perfect accuracy in experiments in friction.
In the present case the agreement is within the ¹/₁₆th of the whole,
and we may regard it as a proof of the law that _the friction of a
pulley is inversely proportional to its circumference_.
166. It is easy to see the reason why friction should diminish when the
size of the pulley is increased. The friction acts at the circumference
of the axle about which the wheel turns; it is there present as a force
tending to retard motion. Now the larger the wheel the greater will be
the distance from the axis at which the force acts which overcomes the
friction, and therefore the less need be the magnitude of the force.
You will perhaps understand this better after the principle of the
lever has been discussed.
167. We may deduce from these considerations the practical maxim
that large pulleys are economical of power. This rule is well known
to engineers; large pulleys should be used, not only for diminishing
friction, but also to avoid loss of power by excessive bending of the
rope. A rope is bent gradually around the circumference of a large
pulley with far less force than is necessary to accommodate it to a
smaller pulley: the rope also is apt to become injured by excessive
bending. In coal pits the trucks laden with coal are hoisted to the
surface by means of wire ropes which pass from the pit over a pulley
into the engine-house: this pulley is of very large dimensions, for the
reasons we have pointed out.