A giant electric shovel, the most human of machines; it digs like a Titan and is used for all kinds of excavating; it is simply a combination of levers and pulleys. The picture shows the largest shovel in the world. The dipper has a capacity of 8 cubic yards (= 10 tons of coal). (Courtesy of the Marion Shovel Company)

Locomotive cranes again merely combinations of levers and pulleys. The upper one is the world's largest wrecking crane; it can be operated by one man; it can lift more than 200 tons; it replaces a crew of 50 men; it is shown here hoisting another locomotive crane. (Courtesy of the Orton Crane and Shovel Company)

Now the perpendicular distances l and l' from the fulcrum to the line of action of the forces are called the lever arms of the forces E and R, and the product of a force by its lever arm is called the moment of that force (Fig. 117). These experiments on the lever may then be generalized in the following law: The moment of the effort is equal to the moment of the resistance. Algebraically stated, it is

It will be seen that the mechanical advantage of the lever, namely R/E, is equal to l/V'; that is, to the lever arm of the

Line of action

of force

FIG. 117. The moment of the

effort divided by the lever arm of force of the rider's foot is the

the resistance.

132. General laws of the lever.

force times the arm, F xl

If parallel forces are applied at several points on a lever, as in Figs. 118 and 119, it will be found, in the particular cases illustrated, that for equilibrium

200 × 30 = 100 x 20 + 100 × 40,

and 300 × 20 + 50 × 40 = 100 × 15 + 200 × 32.5;

that is, the sum of all the moments which are tending to turn the lever in one di

rection about any axis is

40

32.5

B1

50

200

FIG. 118

FIG. 119

Condition of equilibrium of a bar acted upon by several forces

equal to the sum of all the moments tending to turn it in the opposite direction.

If, further, we support the levers of Figs. 118 and 119 by spring balances attached at P, we shall find, after allowing for the weight of the stick, that the two forces indicated by the balances are, respectively, 200 + 100 + 100 = 400 and 300+100+200 - 50 = 550; that is, the sum of all the forces acting in one direction on the lever is equal to the sum of all the forces acting in the opposite direction.

133. The couple. There is one case in which parallel forces can have no single force as their resultant; namely, the case represented in Fig. 120. Such a pair of equal and opposite forces acting at different points on a lever is called a couple and can be neutralized only by another couple tending to produce rotation in the opposite direction. The moment of such a couple is evidently F1 Xoa + F2 X ob = F1 X ab; that is, it is one of the forces times the total distance between them.

E

a

FIG. 120. The couple

R

134. Work expended upon and accomplished by the lever. We have just seen that when the lever is in equilibrium that is, when it is at rest or is moving uniformly- the relation between the effort E and the resistance R is shown in the equation of moments; namely El=Rl'. Let us now suppose, precisely as in the case of the pulleys, that the force E raises the weight R through a small distance s'. To accomplish this the point A to which E is attached must move through a distance s (Fig. 121). From the similarity of the triangles APn and BPm it will be seen that l/l' is equal to s/s'. Hence equation (4), which represents the law of the lever and which may be written E/R = l'/l, may also be written in the form

FIG. 121. Showing that the equation of moments, El Rl', is equivalent

=

to Es= Rs'

Now Es represents the work done by the effort E, and Rs' the work done against the resistance R. Hence the law of moments, which has just been found by experiment to be the law of the lever, is equivalent to the statement that whenever work is accomplished by the use of the lever, the work expended upon the lever by the effort E is equal to the work accomplished by the lever against the resistance R.

135. The three classes of levers. Although the law stated in § 134 applies to all forms of the lever, it is customary to divide them into three classes, as follows:

1. In levers of the first class the fulcrum P is between the acting force E and the resisting force R (Fig. 122). The mechanical advantage of levers of this class is greater or less than unity according as the lever arm I of the effort is greater or less than the lever arm l' of the resistance.

2. In levers of the second class the resistance R is between the effort E and the fulcrum P (Fig. 123). Here the lever arm of the effort, that is, the distance from E to P, is necessarily greater than the lever arm of the resistance, that is, the