Moveable pulley of cast iron 3"·25 diameter, groove
0"·6 wide, wrought iron axle 0"·6 diameter;
fixed pulley of cast iron 5" diameter, groove
0"·4 wide, wrought iron axle 0"·6 diameter,
axles oiled; flexible plaited rope 0"·25
diameter; velocity ratio 2, mechanical efficiency
1·8, useful effect 90 per cent.; formula
_P_ = 2·21 + 0·5453 _R_.
+-----------+---------+----------+----------+------------------+
| | | | P. | Discrepancies |
| Number of | R. | Observed |Calculated| between observed |
|Experiment.| Load | power | power | and calculated |
| | in lbs. | in lbs. | in lbs. | powers. |
+-----------+---------+----------+----------+------------------+
| 1 | 28 | 17·5 | 17·5 | 0·0 |
| 2 | 57 | 33·5 | 33·3 | - 0·2 |
| 3 | 85 | 48·5 | 48·6 | + 0·1 |
| 4 | 113 | 64·0 | 63·8 | - 0·2 |
| 5 | 142 | 80·0 | 79·6 | - 0·4 |
| 6 | 170 | 94·5 | 94·9 | + 0·4 |
| 7 | 198 | 110·5 | 110·2 | - 0·3 |
| 8 | 226 | 125·5 | 125·5 | 0·0 |
+-----------+---------+----------+----------+------------------+
The dimensions of the pulleys are precisely stated because, for
pulleys of different construction, the numerical coefficients would
not necessarily be the same. An attentive study of this table will,
however, show the general character of the relation between the power
and the load in all arrangements of this class.
The table consists of five columns. The first contains merely the
numbers of the experiments for convenience of reference. In the second
column, headed _R_, the loads, expressed in pounds, which are raised in
each experiment, are given; that is, the weight attached to the hook,
not including the weight of the lower pulley. The weight of this pulley
is not included in the stated loads. In the third column the powers are
recorded, which were found to be sufficient to raise the corresponding
loads in the second column. Thus, in experiment 7, it is found that a
power of 110·5 lbs. will be sufficient to raise a load of 198 lbs. The
third column has thus been determined by gradually increasing the power
until motion begins.
195. From an examination of the columns showing the power and the load,
we see that the power always amounts to more than half the load. The
excess is partly due to a small portion of the power (about 1·5 lbs.)
being employed in raising the lower block, and partly to friction. For
example, in experiment 7, if there had been no friction and if the
lower block were without weight, a power of 99 lbs. would have been
sufficient; but, owing to the presence of these disturbing causes,
110·5 lbs. are necessary: of this amount 1·5 lbs. is due to the weight
of the pulley, 10 lbs. is the force of friction, and the remaining 99
lbs. raises the load.
196. By a calculation based on this table we have ascertained a certain
relation between the power and the load; they are connected by the
formula which may be enunciated as follows:
The power is found by multiplying the weight of the load into 0·5453,
and adding 2·21 to the product. Calling _P_ the power and _R_ the load,
we may express the relation thus: _P_ = 2·21 + 0·5453 _R_. For example,
in experiment 5, the product of 142 and 0·5453 is 77·43, to which, when
2·21 is added, we find for _P_ 79·64, very nearly the same as 80 lbs.,
the observed value of the power.
In the fourth column the values of _P_ calculated by means of this
formula are given, and in the last we exhibit the discrepancies between
the observed and the calculated values for the sake of comparison.
It will be seen that the discrepancy in no case amounts to 0·5 lb.,
consequently the formula expresses the experiments very well. The mode
of deducing it is given in the Appendix.
197. The quantity 2·21 is partly that portion of the power expended in
overcoming the weight of the moveable pulley, and partly arises from
friction.
198. We can readily calculate from the formula how much power will
be required to raise a given weight; for example, suppose 200 lbs.
be attached to the moveable pulley, we find that 111 lbs. must be
applied as the power. But in order to raise 200 lbs. one foot, the
power exerted must act over two feet; hence the number of foot-pounds
required is 2 × 111 = 222. The quantity of energy that is lost is 22
foot-pounds. Out of every 222 foot-pounds applied, 200 are usefully
employed; that is to say, about 90 per cent. of the applied energy is
utilized, while the remaining 10 per cent. is lost by friction.