Whence we obtain for the modulus (Art. 152.), observing

R

1

sin. (+3+12) Cos. 1, cos. 12 Cos. $3 2cos. (12-42-3) cos. (+1) sin. (1-2)

THE WEDGE DRIVEN BY PRESSURE.

B

246. Let ACB represent an isosceles wedge, whose angle ACB is represented by 2, and which is driven between the two resisting surfaces DE and DF, by the pressure P1. Let R, and R, represent the resistances of these surfaces upon the acting surfaces CA and CB of the wedge when it is upon the point of moving forwards. Then are the directions of R, and R2 inclined respectively to the perpendiculars Gs and Rt to the faces CA and CB of the wedge, at angles each equal to the limiting angle of resistance 4. The pressures R, and R, are therefore equally inclined to the axis of the wedge, and to the direction of P,, whence it follows that R1 =R, and therefore (Art. 13.) that P1=2R, cos. GOR. Now, since CGOR is a quadrilateral figure, its four angles are equal to two right angles; therefore GOR=27-GCR-OGC

П

ORC. But GCR=2; OGC=ORC=3+4:

2

.:. GOR=

Whence it follows (equation 121) that the modulus of the

wedge is

sin.(+)

U1=U2

sin.

(304).

This equation may be placed under the form

U1=U2 (cot. + cot. } sin. p.

The work lost by reason of the friction of the wedge is greater, therefore, as the angle of the wedge is less; and infinite for a finite value of p, and an infinitely small value of .

B

The angle of the wedge.

247. Let the pressure P1, instead of being that just sufficient to drive the wedge, be now supposed to be that which is only just sufficient to keep it in its place when driven. The two surfaces of the wedge being, under these circumstances, upon the point of sliding backwards upon those between which the wedge is driven, at their points of contact G and R, it is evident that the directions of the resistances iG and iR upon those points, must be inclined. to the normals SG and tR at angles, each equal to the limiting angle of resistance, but measured on the sides of those normals opposite to those on which the re

sistances RG and R,R are applied.*

In order to adapt equation (303) to this case, we have only then to give to a negative value in that equation. It will then become

P1=2R, sin. (-4). . . . . (305).

* This will at once be apparent, if we consider that the direction of the resultant pressure upon the wedge at G must, in the one case, be such, that, if it acted alone, it would cause the surface of the wedge to slip downwards on the surface of the mass at that point, and in the other case upwards; and that the resistance of the mass is in each case opposite to this resultant pressure.

So long as is greater than 4, or the angle C of the wedge greater than twice the limiting angle of resistance, P, is positive; whence it follows that a certain pressure acting in the direction in which the wedge is driven, and represented in amount by the above formula, is, in this case, necessary to keep the wedge from receding from any position into which it has been driven. So that if, in this case, the pressure P1 be wholly removed, or if its value become less than that represented by the above formula, then the wedge will recede from any position into which it has been driven, or it will be started. If be less than, or the angle C of the wedge less than twice the limiting angle of resistance, P, will become negative; so that, in this case, a pressure, opposite in direction to that by which the wedge has been driven, will have become necessary to cause it to recede from the position into which it has been driven; whence it follows, that if the pressure P be now wholly removed, the wedge will remain fixed in that position; and moreover that it will still remain fixed, although a certain pressure be applied to cause it to recede, provided that pressure do not exceed the negative value of P1, determined by the formula.

It is this property of remaining fixed in any position into which it is driven when the force which drives it is removed, that characterises the wedge, and renders it superior to every other implement driven by impact.

It is evidently, therefore, a principle in the formation of a wedge to be thus used, that its angle should be less than twice the limiting angle of resistance between the material which forms its surface, and that of the mass into which it is to be driven.

THE WEDGE DRIVEN BY IMPACT.

248. The wedge is usually driven by the impinging of a heavy body with a greater or less velocity upon its back, in the direction of its axis. Let W represent the weight of such a body, and V its velocity, every element of it being conceived to move with the same velocity. The work ac

cumulated in this body, when it strikes the wedge, will then be

1 W 2 g

V2. Now the whole of this

represented (Art. 66.) by work is done by it upon the wedge, and by the wedge upon the resistances of the surfaces opposed to its motion; if the bodies are supposed to come to rest after the impact, and if the influence of the elasticity and mutual compression of the surfaces of the striking body and of the wedge are neglected, and if no permanent compression of their surfaces follows the 1 WV2

impact.* .. U1 2 9

Substituting this value of U, in equation 303, and solving in respect to Us, we have

by which equation the work U, yielded upon the resistances opposed to the motion of the wedge by the impact of a given

*The influence of these elements on the result may be deduced from the principles about to be laid down in the chapter upon impact. It results from these, that if the surfaces of the impinging body and the back of the wedge, by which the impact is given and received, be exceedingly hard, as compared with the surfaces between which the wedge is driven, then the mutual pressure of the impinging surfaces will be exceedingly great as compared with the resistance opposed to the motion of the wedge. Now, this latter being neglected, as compared with the former, the work received or gained by the wedge from the impact of the hammer will be shown in (1+e) W,WV2

the chapter upon impact to be represented by 2(W1+W), where W, represents the weight of the hammer, W, the weight of the wedge, and e that measure of the elasticity whose value is unity when the elasticity is perfect. Equating this expression with the value of U, (equation 304), and neglecting the effect of the elasticity and compression of the surfaces G and R, between which the wedge is driven, we shall obtain the approximation

From this expression it follows, that the useful work is the greatest, other things being the same, when the weight of the wedge is equal to the weight of the hammer, and when the striking surfaces are hard metals, so that the value of e may approach the nearest possible to unity.

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.

K

H

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 yielded upon the resistances opposed to its motion at either of its faces, is represented by equa

R

2

tions (296) and (297). Supposing therefore this wedge, like the former, to be driven by impact, substituting as before for U1

1 W.

its value V2, and solving in respect to U2, we have in

29

the case in which the face AB of the wedge is its driving surface

9 sin. (1+1+2)

when the base BC of the wedge is its driving surface,

249. If the power of the wedge be applied by the interven

tion of an inclined plane moveable in a direction at right angles to the direction of the impact*, as shown in the accompanying figure, then substituting for U, in equation (300) half the vis viva of the impinging body, and solving, as before, in respect to U,, we have

cos. (+1+3) tan. cos. P

(309).

This is the form under which the power of the wedge is applied for

the expressing of oil.