28 APPLIED MECHANICS 
machine. When the law of a machine is known to be a straight line law 
it is evident that since a line is fixed when two points in it are known, 
the law of the machine can be found from two values of W and the two 
corresponding values of P. 
The quantity c is evidently the magnitude of the effort required to 
drive the machine unloaded. 
The effort to overcome a resistance W when the machine is friction- 
less is P’=W+V, where V is the velocity ratio of the machine. The 
efficiency is obviously equal to the ratio of the effort without friction to 
the effort with friction, hence 
E= te: WwW 1 
W 
on 
As W increases the term nud diminishes, and by making W large enough . 
will become so small that it may be neglected; hence the limit to the 
efficiency of the machine is TV? assuming of course that the law of the 
m 
machine, P=mW +c, remains true with the increased value of W. 
The mechanical advantage ; 
Wc W WwW 1 
Po mW+e a's 
mp 
W 
and reasoning as for the maximum efficiency, it follows that the maximum 
mechanical advantage is ze 
Since M=EV, it follows that if an efficiency curve be drawn this 
curve will also represent the mechanical advantage, but to a scale whose 
unit is 1/V of the unit of the efficiency scale. 
Exercises IIIa. 
1. A man raises a weight of 25 kilogrammes to a height of 22 metres. A 
small steam-engine raises a weight of 98 lbs. to a height of 4 feet, and does the 
same amount of work as the man. Find A, having given that 1 kilogramme 
= 22046 lbs. and 1 metre =39°371 inches. 
2. A block of stone is pulled along level ground at a uniform velocity over a 
distance of 6 yards by a force of 450 lbs. acting on a rope attached to the stone 
and inclined at 45° to the ground. How many foot-pounds of work have been 
done ? 
8. A cylindrical column of granite, 2 feet in diameter and 5 feet long, stands 
with one end on the ground. If the weight of the stone is 170 lbs. per cubic 
foot, how many foot-pounds of work are done in tipping it over into the position 
from which it is about to fall over into a horizontal position ? 
4. A wire rope 250 feet long, and weighing 1 lb. per foot, hangs from the 
drum of a winding-engine, and carries a weight of 15 cwt. at its lower end. 
Find the work done in foot-pounds in winding up the first 200 feet of the rope. 
5. A rectangular tank, 7 feet broad, 8 feet long, and 6 feet deep, is half full of 
water weighing 62'3 lbs. per cubic foot. How many foot-pounds of work will 
be required to raise all the water over the top of the tank? 
6. 250 cubic feet of water are pumped from a rectangular tank, whose base is 
5 feet square, into a cylindrical tank, whose base is 6 feet in diameter. The 
. bottom of the cylindrical tank is 20 feet above the bottom of the other. Find 
the work done in foot-pounds, taking the weight of a cubic foot of water as 
62:3 lbs. 
