30 APPLIED MECHANICS 
here tabulated were obtained, W being the load and P the effort, both in lbs. 
Plot these results. Draw the straight line which most nearly represents the 
relation between P and W, and find its equation in the form P=mW-+c. Draw 
the efficiency curve, and calculate the maximum efficiency. 
21. A crane requires the expenditure of 30 foot-tons of work per second to 
lift 10 tons at the rate of 2 feet per second, and 20 foot-tons per second to lift 
3 tons at the rate of 4 feet per second. Assuming that the law connecting the 
rate at which work is done on the crane (A) with the rate at which the crane 
does work (B) is of the form A=pB+gq, where p and gq are constants, find the 
values of p and g, and use the expression to calculate the efficiency of the crane 
for a load of 5 tons lifted at the rate of 2 feet per second. 
47. Energy.—In mechanics the term energy means capacity for 
doing work. . 
Potential Energy is energy due to the relative position of one body to 
another, or of one part of a body to another part when the two bodies 
or the parts of the same body are under the action of a force or forces 
tending to alter their relative positions. For example, a body which is 
allowed to fall towards the earth may be made to do work ; hence before 
it begins to fall it possesses potential energy, or energy due to its 
position in relation to the earth. A compressed spiral spring has 
potential energy, because if it is allowed to resume its unstrained form 
it can be made to do work. Likewise compressed air possesses potential 
energy. ‘The energy stored in a piece of coal is potential energy, and 
under favourable conditions the atoms of the constituents of the coal and 
the atoms of the oxygen of the air will rush together and produce heat 
which may be converted into work. 
Kinetic Energy is energy due to the motion of a body. A gallon of 
water at rest at a height of 100 feet above the level of the sea possesses 
1000 ft.-lbs. of potential energy, and if this water is allowed to fall 
freely to the level of the sea, without doing work on the way, it will in 
every position of its fall possess 1000 ft.-lbs. of energy, but as it 
descends its potential energy will diminish, and the remainder of the 
1000. ft.-lbs. will be stored in the water as kinetic energy. When the 
gallon of water has fallen 25 feet its potential energy will be reduced to 
750 ft.-lbs., and its kinetic energy will then be 250 ft.-lbs. 
If a body of weight W lbs. falls freely from rest through a height of 
h feet it will then have stored in it Wh foot-lbs. of kinetic energy, and its 
velocity will then be v= ,/2yh feet per second. Hence the kinetic 
2 
energy Wh is equal to 2 It is evident that the kinetic energy of a 
body weighing W lbs., and moving with a velocity of v feet per second, 
2 
will be the same, namely, = , Whatever be the cause of the velocity, 
whether, for example, the cause be the force of gravity, as in a falling 
body, or the force of an explosion, as in a gun. 
48. Kinetic Energy of a Rotating Body.—If an indefinitely small 
body of weight w lbs. be moving with a linear velocity v feet per second 
in a circle of radius r feet, then its angular velocity » in radians per 
; herd Wes se - We Wwr? 
second is equal to v/7, and its kinetic energy is "a9 aR ft.-Ibs. 
If a body of weight W, rotating about a fixed axis with an angular 
velocity w, be divided into indefinitely small parts of weights w,, w,, ts, 
