928

THEORY AND PRACTICE OF ENGINEERING.

Book IJ.

applied to the convex or concave surface of the cylinder, and one of the sides of the prismalways forms an acute angle with the axis of the cylinder, this angle being everywhere the

To raise a weight by the nut and screw, the first may befixed, and the latter left free, or the reverse; in the first instancea lever is applied to the head, which not only turns in the nut,but raises the weight Q attached to its extremity ; but whenthe screw has the nut movable, the power is so applied, thatin turning the nut round the screw, it is elevated, raising atthe same time the weight attached to it, as in the press.

Calling F the power, and Q the resistance to be overcome, itis evident that the first should act in the plane of the move-ment of its lever, or perpendicular to the axis of this move-ment, as in the wheel and axle; and if we abstract thefriction, its labour should be equal to that of the resistanceproduced in the same time. It must be remarked that whenthis latter rises to a certain height, the point of application ofthe power must run over a certain arc, and in virtue of theproperty of the screw, the relation between these two quan-tities is always the same at the end of any given time, what-ever may be the height to which the resistance has mounted,and the arc described by the power, provided that the arc andthe height be simultaneous. We will now consider what takesplace in a complete revolution : the power has described a cir-cumference, whilst the resistance is raised one turn of thescrew. If we call A the height of the turn, and R the distancefrom the point of application of the power of the axis, A and2 P R will be the space described by the resistance Q, and bythe power F in their proper directions; thus F and 2 PR willbe the labour of the power, and QA that of the resistance atthe end of a complete revolution, and we shall have for thecondition of the equilibrium independently of friction,

F x 2 PR= QA, orF : Q : : A : 2 PR,

Fig. 1248.

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Fig. 1249.

which indicates that the power is to the resistance, as the turn of the screw is to thecircumference described by the point of application of the power : so that the power of thescrew increases as the distance between the threads diminishes.

In the endless or perpetual Screw, there will be an _ Q

equilibrium, when the power is to the weight, as thedistance of the threads multiplied by the radius of theaxle, is to the distance of the power from the axis ofthe screw multiplied by the wheels’ radius. SupposeP the power, and x its effect on the toothed-wheel:let R be the radius of the wheel, D the distance be-tween the threads of the screw, r the radius of the axle,

W the weight, and C the circumference described bythe power,

P x C — x x D:

and by the properties of the wheel and axle,xx R = Wxr:multiplying these equalities,

P x C x x x R=x a D x W x r; Fig. 1250.

we then have PxCxR = WxDxr,or, P : W : : D x r : C x R.

The power, then, is to the weight, as the distance between the threads multiplied by theradius of the axle, is to the circumference described by the power multiplied by the radiusof the wheel.

The Lever is of three kinds, and may be called an inflexible bar, capable of moving rounda fixed pivot or fulcrum; its properties are various.

A lever of the first kind has the power andthe weight on opposite sides of the fulcrum, asin the balance, steelyards, scissors, See. Supposingtwo equal weights fixed at the ends of the equalarms of the lever, it is evident there will be a per-fect equilibrium, since there is no reason why eithershould preponderate.

P : W*BC : AC.

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Fig. J25t.