926
THEORY AND l'HA(TU’K OF ENGINEERING.
Book IT.
Fig. 1239
Fig. 1210.
The edge C, called the cuttingedge of the wedge, is the most acuteangle of the triangle A B C. Theface B A is the head, and A C andB C are the sides.
The use of the wedge consists indriving it, by the cutting side C, intothe substance to be split, so thatwhen driven it rests on its sides in twoplaces, a, b. Suppose the force F,which causes it to descend perpen-dicular to the head, (if it were ol>-lique, the wedge would turn,) has thepower of resisting or overcoming thematter to which the wedge is applied,which resistance may be representedas two forces Q and It applied at ab,perpendicular to the respective sides AC and B C. The force F, which produces theequilibrium, will be equal, and directly opposed to their resultant, and as this passes bythe point of concurrence of the forces Q and R, it is necessary that the three forces F, Q,and R should be in the same point, or the wedge would be overthrown. If these re-sistances then be given, the diagonal of the parallelogram constructed on their intensities,and at their point of concurrence, will give the intensity of the force F to be perpen-dicularly applied at the head of the wedge required to overcome these resistances. Takingon the direction of Q, from the point O, a line or distance ()>«, equal to the force, andcarrying through the point m a line perpendicular to B C, as tun, we shall have the tri-angle Own, two sides of which represent the two resistances Q and U, and the third theirresultant or the power F, which is equal or directly opposed to it : the triangle ()w» Uequal to the angle C, the angle m»0 is equal to the angle B, and the angle w()« isequal to the angle A. Thus the sides On and A B, opposed to these angles, are homo-logous ; the same is the case with the sides m O and A C , and the sides m n and B C ;we shall then have the following proportions; On: F
m O : : mn : : A B: AC: B C, or, replacing the dimen-sions On, mO, mn by the forces F, Q, and R, which theyrepresent, F : O : R : : AB : AC: B C, which indicatethat in the equilibrium of the wedge, the perpendicularpower at the head, and the two resistances perpendicularto the sides, are proportioned to that head and these sides ;as when a force is applied to drive it, that force is to theresistance which the parts of the body oppose to it, asthe head of the wedge is to the sum of its sides.
Several forms are given to the wedge; among them isone resembling the triangle ABC, with the rectangle inA, the points of support being at a and b, .and theresistances in the direction Q and R. The powerF, which is perpendicular to ab, cannot concur at thecommon point of the direction of these resistancesu pi ess they arc applied according to the direction ofthe line F6, parallel to the side AC , and passing by thepoint of application of the resistance which is perpen-dicular to the hypotenuse.
In the example where the wedge is an isosceles triangle, which is the common formgiven it, we have A C equal to B C, and, agreeable to what has been already stated, therelations which satisfy the equilibrium are
Q ='XB-’ andIl = TB-
We see, on account of A C being equal to BC, we haveQ,— R, whence we have the proportion ofF : Q : : A B : A C.
When it is required to split bodies, wedges are used,the heads of which are small in proportion to their sides ;if ten times less, as A B is the tenth of A C , F will be thetenth of the reaction, Q, to be overcome ; thus the advantagedrawn from the wedge is, that in diminishing the head, orthe angle of cutting, we can with a small force overcomeconsiderable resistance, as ten times or an hundred times,by making the head a tenth or an hundredth part of its side.