7 >” 
ot WORK AND ENERGY 27 
i enema of the working point when these displacements are in- 
nitely small. 
The ratio of W to P is called the mechanical advantage of the 
machine. 
In all machines a certain amount of work is wasted in overcoming 
friction, and the result is that Pa, the work done by the effort in a given 
time, which may be called the total work or the work put into the 
machine, is greater than Wd, the work done on the resistance in the same 
_ time, which may be called the useful work or the work got out of the 
machine, and the difference between these two quantities of work is the 
lost work. 
_ The ratio of the useful work to the total work is called the efficiency 
of the machine. The efficiency must evidently be always less than unity. 
The reciprocal of the efficiency is called the counter efficiency. If the 
machine be reversed so that W becomes the effort and P the resistance, 
the efficiency under this condition is called the reversed efficiency. 
Let E=efficiency, M=mechanical advantage, and V = velocity ratio, 
WwW Wd M 
then v=5, M= p> E= Pav The lost work is Pa—Whb, and 
~ 
assuming that the lost work is the same when the machine is reversed 
under the action of W as the effort, Wd must be greater than Pa— WA, 
or — must be greater than 3. A machine will therefore not reverse 
under the action of the resistance W unless its efficiency is greater than 
50 per cent. 
46. Usual Relation between the Effort and Resistance in a 
Machine.—If experiments are made with a machine by varying the 
useful resistance W and finding the corresponding values of the effort P, 
it is found that if the results are plotted on squared paper the points 
thus obtained generally lie very nearly in a straight line, and if the 
straight line which most nearly contains all 
the points be drawn, the equation to this line Y 8 
is P=mW +c, where m and ¢ are constants ™m 
for the particular machine. 
In Fig. 26 the dots represent points 
plotted as described above, the values of W ~l~— L 
being measured horizontally from the vertical 
axis OY, and the values of P vertically from 0 N x 
the horizontal axis OX. AB is the straight Fic. 26. 
line which most nearly contains the points. 
Take any point Q in AB, draw QM parallel toOX to meet OY at M, and 
QN parallel to OY to meet OX at N. Then QM and QN represent corre- 
sponding values of W and P respectively. Draw AL parallel to OX to meet 
QN at L. It is evident that wherever Q may be taken in AB the ratio 
QL+AL will be the same. Let QL+AL=™m, and let OA=c, then 
_QL_QN-LN P-c £ 
m= 77 = a a therefore P=mW +e. 
In plotting it is not necessary that the scale for P be the same as 
that for W, but in determining the value of m care must be taken to_ 
measure QL and OA with the scale for P, and AL with the scale for W. 
The relation between P and W is sometimes called the law of the 
