THE LEVER.] 



MECHANICAL PHILOSOPHY. STATICS. 



719 



its vertex and the centr^rf gravity of the base, at a 

 distance equal to threej^fchs of its length from the 

 former, or one-fourth oj^Hrength from the latter. 



The above reasoning S^Rtogether independent of the 

 number of the sides of the polygon, and since we may 

 conceive a curve to be made up of an infinite number of 

 straight lines, or a polygon bounded by an infinite 

 number of small sides, the above solution enables us to 



determine the centre of gravity of a cone with a curvi- 

 linear base. 



Hence the centre of gravity of a right or oblique cone 

 on any curvilinear base is found by joining the centre 

 of gravity of the base with the apex of the cone, and 

 taking a point in this line, equal to one-fourth of its 

 length, measuring this point from the centre of gravity 

 of the base. 



CHAPTER II. 

 THE MECHANICAL POWERS. 



ANT instrument by means of which a force is communi- 

 cated from one point to another, so as to keep at rest or 

 set in motion a body acted on by another force, is called 

 a macliine. The simplest of these instruments are cords, 

 rods, and hard planet ; and these, by their combinations, 

 form all complex machines, however various their forms 

 and actions. 



In statics, we only consider machines when the forces 

 acting upon them are in a state of equilibrium ; motion 

 will be produced by an addition to some of the forces 

 which produce equilibrium. But the discussion of this 

 branch of the subject belongs to Dynamics, which will 

 be investigated in the following chapter. 



For tho sake of simplicity, and to enable us to apply 

 the theoretical principles we have already proved, we 

 consider cords as being destitute of weight and perfectly 

 flexible, rods and planet as perfectly rigid, inflexible, 

 and without weight. When necessary, we can take the 

 weights of the elementary parts of our machines into 

 consideration, by considering their weights as a force 

 proportional to their weight applied to the centre of 

 gravity, in a vertical direction. 



We also neglect the friction of all surfaces in contact 

 with each other. It is evident, therefore, that our 

 machines will be theoretical ones, which cannot exist in 

 nature, and whose properties cannot be strictly proved 

 by experiment. But if we determine the rigidity, flexi- 

 bility, and friction of the substances composing our 

 machine by experiment, and compare the force exerted 

 by a real machine with the force it ought to exert 

 by theory, we may arrive at a knowledge of the retard- 

 ing forces produced by friction or want of flexibility ; 

 and thus by our theoretical knowledge of the combina- 

 tion of machines, estimate the forces produced by any 

 actual complex machine, the friction or flexibility of 

 whose elements we have determined.* 



MECHANICAL POWERS. The simplest combina- 

 tions of these machines are called the mechanical powers. 

 They are usually regarded as seven in number : 1, the 

 lever ; 2, the wheel and axle ; 3, the toothed wheel ; 4, 

 the pulley ; 6, the inclined plane ; C, the wedge ; 7, the 

 crew. 



The wheel and axle, toothed wheel and pulley, may 

 be regarded as modifications of the lever, and the 

 wedge and screw as particular cases of the inclined 

 plane. 



THE LEVER. The simplest form of a lever is a 

 straight rod, supposed to be inflexible and without 

 weight, resting on a fixed point, somewhere in its length, 

 about which it can turn freely, and having two forces 

 applied at two other points of the rod. 



The fixed point on which it rests, and about which it 

 can turn, is called the fulcrum ; one of the forces applied 

 to it is called the power, and the other the weight. The 

 distances of the points of application of the power and 

 weight, from the fulcrum, are called the arms of the 

 lever. 



There are three kinds of levers, distinguished by the 

 relative position of the power, weight, and fulcrum. 



Lever of the first kind. In the lever of the first 

 kind the power P (Fig. 90) and the weight W act in 

 the same direction on opposite sides of the fulcrum 

 DM Prantrt, Tttuion, and fUzMe Cordt, p. 693. 



Fig. 90. 



F. A crow-bar (Fig. 91), by means of which a man raises 

 a heavy body W by 

 placing one extremity 

 B under W, and rest- 

 ing it on any body 

 C while he presses the 

 extremity B', is an in- 

 stance of a lever of 

 the first kind. 



A poker is another. 

 In this case the coals 

 form the weight, the 

 bar of the grate the 



fulcrum, and the hand tho power. 



The spade is a lever ; the ground against which 

 Fig. 91. 



it is 



pressed when the handle is depressed, in order to turn up 

 the earth in front of it, being the fulcrum. 



Scissors and carpenters' pincers are examples of double 

 levers of the first kind. 



Lever of the second kind. In the lever of the second 

 kind the power P (Fig. 92) and the weight W act in 

 opposite directions on the same side of the fulcrum F, 

 the weight being nearer to the fulcrum than the power. 

 Fig. 93. A cutting-knife (Fig. 



93) and an oar are in- 

 stances of levers of the 

 second kind. In the case 

 of the oar, the arm of 

 the rower is the power, 

 the pressure of the oar 

 on the side of the boat 

 is the weight, and the 

 point of the blade of the 

 oar, which is for a 

 moment stationary in tho 

 water, forms the fulcrum. 

 Nutcrackers give a 



the second kind. 



good illustration 

 a double lever 



of 

 of 



Fig. 93. 



