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133. 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 expressed 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. 122). From the similarity of the triangles APn

R

FIG. 122. Showing that the equation of moments, El = Rľ, is equivalent to Es Rs'

=

and BPm it will be seen that 7/1' is equal to s/s'. Hence equation (4), which represents the law of the lever, and which may be written E/R='/l, may also be written in the form

E/Rs/s, or 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.

134. The three classes of levers. Although the law stated in § 133 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. 123). The mechanical advantage of levers of this class is greater or less than unity according as the lever arm 7 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. 124). 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

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distance from R to P. Hence the mechanical advantage of levers of the second class is always greater than 1.

3. In levers of the third class the acting force is between the resisting force and the fulcrum (Fig. 125). The mechanical advantage is then obviously less than 1, that is, in this type of lever force is always sacrificed for the sake of gaining speed.

QUESTIONS AND PROBLEMS

1. In which of the three classes of levers does the wheelbarrow belong? grocer's scales? pliers? sugar tongs? a claw hammer? a pump handle?

2. Explain the principle of weighing by the steelyards (Fig. 126). What must be the weight of the bob P if at a distance of 40 cm. from the fulcrum O it balances a weight of 10 kg. placed at a distance of 2 cm. from (?

3. If you knew your own weight, how could you determine the weight of a companion if you had only a teeter board and a foot rule? 4. How would you arrange a crowbar to use it as a lever of the first class in overturning a heavy object? as a lever of the second class?

5. Why do tinners' shears have long handles and short blades and tailors' shears just the opposite?

6. By reference to moments explain (a) why a door can be closed more easily by pushing at the knob than at a point close to the hinges; (b) why a heavier load can be lifted on a wheelbarrow having long handles than on one with short han

dles; (c) why a long-handled shovel generally has a smaller blade than one with a shorter handle.

7. Two boys carry a load of 60 lb. on a pole between them. If the load is 4 ft. from one boy and 6 ft. from the other, how many pounds does each boy carry? (Consider the force exerted by one of the boys as the effort, the load as the resistance, and the second boy as the fulcrum.)

P

W

FIG. 126. Steelyards

8. Where must a load of 100 lb. be placed on a stick 10 ft. long if the man who holds one end is to support 30 lb. while the man at the other end supports 70 lb.?

9. One end of a piano must be raised to remove a broken caster. The force required is 240 lb. Make a diagram to show how a 6-foot steel bar may be used as a second-class lever to raise the piano with an effort of 40 lb.

10. When a load is carried on a stick over the shoulder, why does the pressure on the shoulder become greater as the load is moved farther out on the stick?

11. A safety valve and weight are arranged as in Fig. 127. If ab is 11⁄2 in. and be 101⁄2 in., what effective steam pressure per square inch is required on the valve to unseat it, if the area of the valve is and the weight of the ball 4 lb.?

12. The diameters of the piston and cylinder of a hydraulic press are respectively 3 in. and 30 in. The piston rod is attached 2 ft. from the fulcrum of a lever 12 ft. long (Fig. 12, p. 17). What force must be applied at the end of the lever to make the press exert a force of 5000 lb. ?

a

ט

FIG. 127

sq. in.

13. Three boys sit on a seesaw as follows: A (= 75 lb.), 4 ft. to the right of the fulcrum; B (= 100 lb.), 7 ft. to the right of the fulcrum; C (= x lb.), 7 ft. to the left of the fulcrum. Equilibrium is produced by a man, 12 ft. to the right of the fulcrum, pushing up with a force of 25 lb. Find C's weight.

THE PRINCIPLE OF WORK

135. Statement of the principle of work. The study of pulleys led us to the conclusion that in all cases where such machines are used the work done by the effort is equal to the work done against the resistance, provided always that friction may be neglected and that the motions are uniform so that none of the force exerted is used in overcoming inertia. The study of levers led to precisely the same result. In Chapter II the study of the hydraulic press showed that the same law applied in this case also, for it was shown that the force on the small piston times the distance through which it moved was equal to the force on the large piston times the distance through which it moved. Similar experiments upon all sorts of machines have shown that the following is an absolutely general law: In all mechanical devices of whatever sort, in all cases where friction may be neglected, the work expended upon the machine is equal to the work accomplished by it.

This important generalization, called "the principle of work," was first stated by Newton in 1687. It has proved to be one of the most fruitful principles ever put forward in the history of physics. By its application it is easy to deduce the relation between the force applied and the force overcome in any sort of machine, provided only that friction is negligible and that the motions take place slowly. It is only necessary to produce, or imagine, a displacement at one end of the machine, and then to measure or calculate the corresponding displacement at the other end. The ratio of the second displacement to the first is the ratio of the force acting to the force overcome.

136. The wheel and axle. Let us apply the work principle to discover the law of the wheel and axle (Fig. 128). When the large wheel has made one revolution, the point A on the rope

moves down a distance equal to the circumference of the wheel. During this time the weight R is lifted a distance equal to the circumference of the axle. Hence the equa

=

w

tion Es Rs becomes Ex 2 TR=RX 2πr where R and r are the radii of the wheel and axle respectively. This equation may be written in the form

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that is, the weight lifted on the axle is as many times the force applied to the wheel as the radius of the wheel is times the radius of the axle.

FIG. 129. The capstan

Otherwise stated,

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the mechanical advantage of the wheel and axle is equal to the radius of the wheel divided by the radius of the axle.

The capstan (Fig. 129) is a special case of the wheel and axle, the length of the lever arm taking the place of the radius of the wheel, and the radius of the barrel corresponding to the radius of the axle.

137. The work principle applied to the inclined plane. The work done against gravity in lifting a weight R (Fig. 130)

from the bottom to the top of a

plane is evidently equal to R times the height h of the plane. But the work done by the acting force E while the carriage of weight Ris being pulled from the bottom to the top of the plane is equal to E times the length of the plane. Hence the principle of

work gives

FIG. 130. The inclined plane

El-Rh, or R/E=1/h;

(6)

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