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=

TOS the inclination of the faces of the plane B to oue another—

.. R,OR,= (~ — 9,) + (5) — 9.)—(1, −1,)—~—(P, +P2)—(1, −1).

';) — :)

Also P,QR‚———R,QM=;-e

Let P,O be produced to V; therefore P,OR,=-R1OV=

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R,QR, = OQM+MQR,. Now, MQR,=,; also, OQM=

T

=

◄—QOV=~~(QOT+TOV)=-— { (5 −9.) + 1 } = 3 − +ïm

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2

sin. (+1, +9.). sin. {3-(4-P.-P.,)}

sin. {(9, +92) + (', —1,)} cos. P,
cos. (,+,) cos. ',-(P,+P,)}

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. (301).

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

that = (0)

sin. (-)

COS. COS.

U1=U,

sin. (9, +9+4,-1) cos. 4, cos. 4, cos. 9,
cos.(4,-,-,) cos.(4, +,) sin. (4-4)

(302).

321

THE WEDGE DRIVEN BY PRESSURE.

246. Let ACB represent an isosceles wedge, whose angle ACB is represented by 24, 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 R, inclined respectively to the perpendicular Gs and Rt to the faces CA and CB of the wedge, at angles cach equal to the limiting angle of resistance. 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 R, R, and therefore (Art. 13 ) that P,=2R, cos. ¿GOR. Now, since CGOR is a quadrilateral figure, its four angles are equal to four right angles; therefore GOR=2π-GCR

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A

T

= +:

OGC-ORC. But GCR=21; OGC=ORC: = 2

..GOR=«—(21+2p) :.JGOR=5−(1+4).

:. P‚=2R, sin. (1+¢) . . . . . (303).

....

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

wedge is

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This equation may be placed under the form

U1=U, {cot. +cot. } sin. .

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 o, and an infinitely small value

of .

The angle of the wedge.

247. Let the pressure P1, instead of being that just suli

B

2

t

cient 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 R upon those points, must be inclined to the normals SG and R at angles, each equal to the limiting angle of resistance, but measured on the sides of those normals opposite to those on which the resistances R,G and RR 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

A

P, 2R, sin. (-9). . . . . (305).

So long as is greater than o, 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 pre-sure P, 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. It be less than 9, 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 P, determined by the formula.

*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.

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.

1 W.

2 g

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 accumulated in this body, when it strikes the wedge, will then be represented (Art. 66.) by V. Now the whole of this 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 1 WV2 follows the impact.

*

.. U1

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2

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* 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 the chapter upon impact to be represented by (1+e)2W1°W2 V2 where W represents the 29(W1+W2)2

1

weight of the hammer, W, the weight of the wedge, and 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

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Substituting this value of U, in equation 304, and solving in respect to U,, we have

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

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 equations (296) and (297). Supposing therefore this wedge, like the former, to be

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F

D
C

driven by impact, substituting as before for U, its value

1 W.

2 g

V', and solving in respect to U,, we have in the case in

which the face AB of the wedge is its driving surface

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

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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.

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