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weight W with a given velocity V is determined; or the weight W necessary to yield a given amount of work when moving with a given velocity; or, lastly, the velocity V with which a body of given weight must impinge to yield a given amount of work.
If the wedge, instead of being isosceles, be of the form of a right angled triangle, as shown in the accompanying figure, the relation between the work U[ done upon its back, and that if yielded upon the resistances opposed to its motion at either of its faces, is represented by equations (296) and (297). Supposing therefore this wedge, like the former, to be driven by impact, substituting as before for U,
1 W .
its value jj —V, and solving in respect to U2, we have in
* 9 the case in which the face AB of the wedge is its driving surface
"2 g 'sin. (i + ft + fr) when the base BC of the wedge is its driving surface,
249. If the power of the wedge be applied by the intervention of an inclined plane moveable in a direction at right angles to the direction of the impact*, as shown in the accompanying figure, then substitutingfor U, in equation (300) half the vis viva of the impinging body, and solving, as before, in respect to U2, we have
* This is the form under which the power of the wedge is applied for the expressing of oil.
If instead of the base of the plane being parallel to the direction of impact, it be inclined to it, as shown in the accompanying figure, then, substituting as above in equation 302, we have
U =l WV' cos. 0, — to-fa) cos. Qi+jgO sin, ('i-'s) (3W) 2 H g 'sin. (Pi+Pa+i! — ia) cos. ix cos. ^ cos. p3"' '*
The Mean Pressure Of Impact.
250. It is evident from equations 306, 307, 308, that, since, whatever may be the weight of the impinging body or the velocity of the impact, a certain finite amount of work Ua is yielded upon the resistances opposed to the motion of the wedge; there is in every such case a certain mean resistance R overcome through a certain space S, in the direction in which that resistance acts, which resistance and space are such, that
RS = U2, and therefore R= ^2.
If therefore the space S be exceedingly small as compared with Uj, there will be an exceedingly great resistance R overcome by the impact through that small space, however slight the impact From this fact the enormous amount of the resistances which the wedge, when struck by the hammer, is made to overcome, is accounted for. The power of thus subduing enormous resistances by impact is not however peculiar to the wedge, it is common to all implements of impact, and belongs to its nature; its effects are rendered permanent in the wedge by the property possessed by that implement of retaining permanently any position into which it is driven between two resisting surfaces, and thereby opposing itself effectually to the tendency of those surfaces, by reason of their elasticity, to recover their original form and position. It is equally true of any the slightest direct
impact of the hammer as of its impact applied through the wedge, that it is sufficient to cause any finite resistance opposed to it to yield through a certain finite space, however great that resistance may be. The difference lies in this, that the surface yielding through this exceedingly small but finite space under the blow of the hammer, immediately recovers itself after the blow if the limits of elasticity be not passed; whereas the space which the wedge is, by such an impact made to traverse, in the direction of its length, becomes a permanent separation.
251. Let the system of two moveable inclined planes represented in fig. p. 345. be formed of exceedingly thin and pliable laminsB, and conceive one of them, A for instance, to be wound upon a convex cylindrical surface, as shown in the accompanying figure, and the
i^5""-^ other, B, upon a concave cylindrical surface having an equal diameter, and the same axis with the other; then will the surfaces EF and GH of these planes represent truly the threads or helices of two screws, one of them of the form called the male screw, and the other the female screw. Let the helix EF be continued, so as to form more than one spire or convolution of the thread; if, then, the cylinder which carries this helix be made to revolve upon its axis by the action of a pressure P, applied to its circumference, and the cylinder which carries the helix GH be prevented from revolving upon its axis by the opposition of an obstacle D, which leaves that cylinder nevertheless free to move in a direction parallel to its axis, it is evident that the helix EF will be made to slide beneath GH, and the cylinder which carries the latter helix to traverse longitudinally; moreover, that the conditions of this mutual action of the helical surfaces EF and GH will be precisely analogous to those of the surfaces of contact of the two moveable inclined planes discussed in Art 244. So that the
conditions of the equilibrium of the pressures P, and P2 in the state bordering upon motion, and the modulus of the system, will be the same in the one case as in the other; with this single exception, that the resistance Ra of the mass on which the plane A rests (see fig. p. 345.) is not, in the case of the screw, applied only to the thin edge of the base of the lamina A, but to the whole extremity of the solid cylinder on which it is fixed, or to a circular projection from that extremity serving it as a pivot. Now if, in equation 299, we assume p2=0, we shall obtain that relation of the pressures P, and P2 in their state bordering upon motion, which would obtain if there were no friction of the extremity of the cylinder on the mass on which it rests; and observing that the pressure P3 is precisely that by which the pivot at the extremity of the cylinder is pressed upon this mass, and therefore the moment (see Art. 175, equation 183) of the resistance to the rotation of the cylinder produced by the
2 friction of this pivot by ^Prf tan. <p.t, where p represents the
radius of the pivot; observing, moreover, that the pressure which must be applied at the circumference of the cylinder to overcome this resistance, above that which would be required to give motion to the screw if there were no such
friction, is represented by ^ P2 - tan.p2, r being taken to re
present the radius of the cylinder, we obtain for the entire value of the pressure P, in the state bordering upon motion,
1 "cos. (i + <p2 4- <p3) o r rThe pressure Pj has here been supposed to be applied to turn the screw at its circumference; it is customary, however, to apply it at some distance from its circumference by the intervention of an arm. If a represent the length of such an arm, measuring from the axis of the cylinder, it is evident that the pressure P, applied to the extremity of that arm, would produce at the circumference of the cylinder a pressure
represented by P,-, which expression being substituted for P, in the preceding equation, and that equation solved in respect to Pp we obtain finally for the relation between P, and P2 in their state bordering upon motion,
1 2W 1 cos. (» + ft + ft) 3\r/ ra J v y
If in like manner we assume in the modulus (equation 300)
ft = 0, and thus determine a relation between the work done
at the driving point and that yielded at the working point,
on the supposition that no work is expended on the friction
of the pivot; and if to the value of L^ thus obtained we add
the work expended upon the resistance of the pivot which is
shown (equation 184) to be represented at each revolution
4 by oWpPj tan. ft, and therefore during n revolutions by
-irnp¥2, we shall obtain the following general expression for
the modulus; the whole expenditure of work due to the prejudicial resistances being taken into account.
sin. (i + ft)cos. ft, 4 _
Ui=IJ, • , \„!'^f +l*npV3 tan. ft
1 ■ cos. (i + ft + ft) tan i 3 r T
Representing by A. the common distance between the threads of the screw, i. e. the space which the nut B is made to traverse at each revolution of the screw; and observ
4 4 U
ing that «xP2=U2, so that ^T»pP2tan.ft=-7r—p tan. ft =
2 %tr p TT . , . 2*r
„ -— . - . U2 tan. ft, in which expression —— = cot. i, we
obtain finally for the modulus of the screw
sin. (i + ft)cos, ft 2p I ,„„
v, P' , ^ X +t,- tan. ft J-cot. i .... (312).
cos. 0 + ft + ft) 3 r r* J >
It is evidently immaterial to the result at what distance from the axis the obstacle D is opposed to the revolution of that cylinder which carries the lamina B; since the amount of that resistance does not enter into the result as expressed in the above formula, but only its direction determined by the angle ft, which angle depends upon the nature of the