WEDGE, in Mechanics, is a prism of wood or metal whose base is a triangle : it is employed to remove two objects from one another laterally, or to rend asunder the parts of a body ; an edge which is parallel to the geometrical axis of the prism being introduced between the objects, or parts of the body, and the whole wedge being then driven forward by a percussive force, as the stroke of a hammer. The nature of percussive force, and of the resistance which a yielding material opposes to the motion of the wedge being, however, imper fectly known, it is usual to consider the motive-power and the resistance as simple pressures, or weights, in estimating the conditions of equilibrium when a wedge is employed as a inpehine.
Let Ares represent the section of a wedge perpendicularly to the mathematical axis; and for simplicity, suppose this section to bo an isosceles triangle. Tho plane passing though AB, perpendicular to A no and to the paper, is called the head or back of the wedge : the planes passing through A0 and no perpendicular to the paper are called tho sides ; and Choir line of section, passing through is, is called the edge.
The motive-power is applied to the head of the wedge; and may be supposed to act in the direction ma perpendicularly to that plane, and passing through c in the edge. Let the material which is to be rent asunder be in contact with the sides of the wedge in lines passing through a and b perpendicularly to the paper; and let the two parts yield as if they were capable of turning about some point c in the direction of 310, produced if necessary : then it is evident that the effect of the wedge will be the same as if a section of the latter, perpendicularly to the axis, were abc. Imagine ad, bd to be drawn perpendicularly to ac and be ; then those lines will meet at a certain point, as d, in the line is a : imagine also the parallelogram adbD of forces to be con structed; then do or 2dN will represent the motive force, and da or db tho pressure which that force exerts at a or b perpendicularly to ac and be. Let r represent the motive force, and a the pressure at a or b; then, in a state of equilibrium, the latter will represent the reaction of the material in tho direction ad or bd ; and we have But the triangle reed Is similar to CX41; therefore dN : da : : ON : ne, uxi consequently r : n : : 2ae (mab) : The position of the point c where the separation of the material is mel'escel to take place varies for different materials, and can only be tiinatertl, or found from experiment : if it were supposed to coiucide sith c, we should have r : 2ex : ac ; rir by similar triangles abc, rtn::e.triece
If the force of the wedge at the points n or b were to be estimated in the direction No or Nb; as when it is required, neglecting friction. to find the force with which ft given pressure r in the direction arc, on the head of the wedge, would make a body at a or b slide in a direction perpendicular to etc : then, representing that force, and c coinciding with c, or ad being now perpendicular to ac, we should have P : 2dN : OT 118 2as to No, or t1.9 AB tO EC.
The point c stall coinciding with c, let 0 represent the angle ACE, or half the angle of the wedge : then en = sin 0, or 7 - 2 sin 0; a also AD : EC : : 2 Sill 0 : COS B : It follows, radius being unity, that r : : : 2 sin : 1, or r =2R sin 0; also r n' : : 2 sin 0 : cos 0, or r = 2 n' tan 0 ; where n and n' represent the pressures perpendicular to A c and to EC respectively.
If it were required to find an equation for the motion of a wedge when acted upon by a force of percussion, a process corresponding to that which fellows must be employed.
Let, as before, ADC be a section of the wedge, which may bo sup posed of iron : let it be introduced between the parts of a body which can yield only in a lateral direction ; and lot it be driven by a mass of iron falling upon it from a point at some given height above. Both the wedge and the hammer, or falling body, must be tinderstood to be elastic ; and it will be convenient to represent the latter by a parallelopiped of iron whose base is equal to the rectangular head of the wedge : let the height of such solid be represented by r, and let the space through which it is supposed to fall be represented by 11 ; then, by dynamics, 2gh will be equal to the square of the velocity of impact. In like manner, let the friction of the wedge, estimated in a direction parallel to CU, be represented by the weight of a parallelo piped of iron whose base is the rectangular head of the wedge; and let the height of such parallelopiped be represented by Q.