=120. The Simple Machines.=--There are but six _simple machines_. All the varieties of machines known are simply modifications and combinations of the six simple machines. The six simple machines are more easily remembered if we separate them into two groups of three each. The first or _lever_ group consists of those machines in which a part revolves about a fixed axis. It contains the _lever_, _pulley_ and _wheel and axle_. The second or _inclined plane group_ includes those having a sloping surface. It contains the _inclined plane_, the _wedge_, and the _screw_.

=121. The Lever.=--The _lever_ is one of the simple machines most frequently used, being seen in scissors, broom, coal shovel, whip, wheelbarrow, tongs, etc. _The lever consists of a rigid bar capable of turning about a fixed axis called the fulcrum._ In studying a lever, one wishes to know what weight or resistance it can overcome when a certain force is applied to it. Diagrams of levers, therefore, contain the letters _w_ and _f_. In addition to these, _O_ stands for the fulcrum on which it turns. By referring to Fig. 86, _a_, _b_, _c_, one may notice that each of these may occupy the middle position between the other two. The two forces (other than the one exerted by the fulcrum) acting on a lever always oppose each other in the matter of changing rotation. They may be considered as a pair of parallel forces acting on a body, each tending to produce rotation.

[Ill.u.s.tration: FIG. 86.--The three cla.s.ses of levers.]

=122. Moment of Force.=--The _effectiveness_ of each force may therefore be determined by computing its _moment_ about the fixed axis (see Art.

84), that is, by multiplying each force by its distance to the fulcrum or axis of rotation. Let a meter stick have a small hole bored through it at the 50 cm. mark near one edge, and let it be mounted on a nail driven into a vertical support and balanced by sliding a bent wire along it. Suspend by a fine wire or thread a 100 g. weight, 15 cm. from the nail and a 50 g. weight 30 cm. from the nail, on the other side of the support. These two weights will be found to balance. When viewed from this side _A_ (Fig. 87) tends to turn the lever in a clockwise direction (down at right), _B_ in the counter-clockwise direction (down at left).

Since the lever balances, the forces have equal and opposite effects in changing its rotation as may also be computed by determining the moment of each force by multiplying each by its distance from the fulcrum.

Therefore the _effectiveness_ of a force in changing rotation depends upon the distance from it to the axis as well as upon the magnitude of the force.

[Ill.u.s.tration: FIG. 87.--The two moments are equal about _C_. 100 15 = 50 30.]

From the experiment just described, the moment of the acting force equals the moment of the weight or _f D_{f} = w D_{w}_, or the effort times the effort arm equals the weight times the weight arm. This equation is called the law of the lever. It corresponds to the general law of machines and may also be written _w: f = D_{f}: D_{w}_.

=123. Mechanical Advantage.=--A lever often gives an advantage because by its use one may lift a stone or weight which the unaided strength of man could not move. If the lever is used in lifting a stone weighing 500 lbs., the force available being only 100 lbs., then its _mechanical advantage_ would be 5, the ratio of _w:f_. In a similar way, the mechanical advantage of any machine is found by finding the ratio of the resistance or weight to the effort. What must be the relative lengths of the effort arm and resistance or weight arm in the example just mentioned? Since the effort times the effort arm equals the weight times the weight arm, if _f D_{f} = w D_{w}_, then _D_{f}_ is five times _D_{w}_. Hence the mechanical advantage of a lever is easily found by finding the ratio of the effort arm to the weight arm.

Important Topics

1. Advantage of machines.

2. Machines cannot create energy.

3. Law of machines.

4. Six simple machines.

5. Lever and principle of moments.

6. Mechanical advantage of a machine.

Exercises

1. Give six examples of levers you use.

2. Fig. 88_a_ represents a pair of paper shears, 88_b_ a pair of tinner's shears. Which has the greater mechanical advantage? Why?

Explain why each has the most effective shape for its particular work.

[Ill.u.s.tration: FIG. 88.--(_a_) Paper shears. (_b_) Tinner's shears.]

3. Find examples of levers in a sewing machine.

4. What would result if, in Art. 122, the 100 g. weight were put 25 cm.

from O and the 50 g. weight 45 cm. from O? Why? Explain using principle of moments.

5. How is the lever principle applied in rowing a boat?

6. When you cut cardboard with shears, why do you open them wide and cut near the pivot?

7. In carrying a load on a stick over the shoulder should the pack be carried near the shoulder or out on the stick? Why?

8. How can two boys on a see-saw start it without touching the ground?

9. In lifting a shovel full of sand do you lift up with one hand as hard as you push down with the other? Why?

[Ill.u.s.tration: FIG. 89.--The hammer is a bent lever. What is its mechanical advantage?]

10. Why must the hinges of a gate 3 ft. high and 16 ft. wide be stronger than the hinges of a gate 16 ft. high and 3 ft. wide?

11. When one sweeps with a broom do the hands do equal amounts of work?

Explain.

12. A bar 6 ft. long is used as a lever to lift a weight of 500 lbs. If the fulcrum is placed 6 in. from the weight, what will be the effort required? Note: two arrangements of weight, fulcrum and effort are possible.

13. The handle of a hammer is 12 in. long and the claw that is used in drawing a nail is 2.5 in. long. (See Fig. 89.) A force of 25 lbs. is required to draw the nail. What is the resistance of the nail?

14. The effective length of the head of a hammer is 2 in. The handle is 15 in. long and the nail holds in the wood with a force of 500 lbs. Only 60 lbs. of force is available at the end of the handle. What will be the result?

15. If an effort of 50 lbs. acting on a machine moves 10 ft., how far can it lift a weight of 1000 lbs.?

16. A bar 10 ft. long is to be used as a lever. The weight is kept 2 ft.

from the fulcrum. What different levers can it represent?

17. The effort arm of a lever is 6 ft., the weight arm 6 in. How long will the lever be? Give all possible answers.

18. Two boys carry a weight of 100 lbs. on a pole 5 ft. long between them. Where should the weight be placed in order that one boy may carry one and one-fourth times as much as the other?

(4) THE WHEEL AND AXLE AND THE PULLEY

=124. The Wheel and Axle.=--1. One of the simple machines most commonly applied in compound machines is the _wheel_ and _axle_. It consists of a wheel _H_ mounted on a cylinder _Y_ so fastened together that both turn on the same axis. In Fig. 90, ropes are shown attached to the circ.u.mferences of the wheel and axle. Sometimes a hand wheel is used as on the brake of a freight or street car, or simply a crank and handle is used, as in Fig. 91. The _capstan_ is used in moving buildings.

Sometimes two or three wheels and axles are geared together as on a derrick or crane as in Fig. 92.

[Ill.u.s.tration: FIG. 90.--The wheel and axle.]

[Ill.u.s.tration: FIG. 91.--Windla.s.s used in drawing water from a well.]

[Ill.u.s.tration: FIG. 92.--A portable crane.]

[Ill.u.s.tration: FIG. 93.--The wheel and axle considered as a lever.]

[Ill.u.s.tration: FIG. 94.--View of transmission gears in an automobile. 1, Drive gear; 2, High and intermediate gear; 3, Low and reverse gear; 4, 8, Reverse idler gears; 5, 6, 7, Countershaft gears. (_Courtesy of the Automobile Journal_.)]

[Ill.u.s.tration: FIG. 95.--Reducing gear of a steam turbine.]

Fig. 93 is a diagram showing that the wheel and axle acts like a lever.

The axis _D_ is the fulcrum, the effort is applied at _F_, at the extremity of a radius of the wheel and the resisting weight _W_ at the extremity of a radius of the axle. Hence, if _D_{f}_, the effort distance, is three times _D_{w}_, the weight distance, the weight that can be supported is three times the effort. Here as in the lever, _f D_{f} = w D_{w}_, or _w:f = D_{f}:D_{w}_, or _the ratio of the weight to the effort equals the ratio of the radius of the wheel to the radius of the axle_. This is therefore the mechanical advantage of the wheel and axle. Since the diameters or circ.u.mferences are in the same ratio as the radii these can be used instead of the radii. Sometimes, when _increased speed_ instead of increased force is desired, the radius of the wheel or part to which power is applied is less than that of the axle. This is seen in the bicycle, buzzsaw, and blower. Sometimes geared wheels using the principle of the wheel and axle are used to reduce speed, as in the _transmission_ of an automobile (see Fig. 94), or the reducing gear of a steam turbine. (See Figs. 95 and 293.)