WEDGE AND SCREW.] 



MECHANICAL PHILOSOPHY. ST ATI CS. 



731 



THE INCLINED PLANE. The inclined plane is 

 the fifth mechanical power. It consists of a plane sur- 

 face A B, which is supposed to be perfect in hardness 

 and smoothness, and inclined at some angle a to the 

 horizontal line. A heavy body, whose weight is W (Fig. 

 133 1), resting on the plane, and supported by a force 

 P acting in some direction D E constitutes the weight 

 in this machine. 



fig. 133-(1). 



Fig. 13*-(2). 



Let ft be the angle D E makes with B A, and B C be 

 drawn at right angles to A C : a is called the inclination 

 of the plane, B C its 

 height, A B its length, 

 and A C its base. 

 When tlicre is e<iui- 

 librium, D will bo 

 acted on by three 

 forces, the force P 

 acting in the direction 

 DE, the weight W 

 acting vertically down- 

 wards, and the un- 

 known reaction R pro- 

 duced by the pressure 

 of the body D on the 

 plaue AB, which acts 

 perpendicularly to the 

 surface of the plane. 



Let R, P, and W 

 bo represented ill mag- 

 nitude and direction by D R, I) P, and D W (2). 



Resolving these forces parallel aiid perpendicular to 

 the surface of the plane A B, as shown in Fig. 2, we 

 have the following conditions of equilibrium : 



W sin. a = P cos. ft, or 



W cos. ft 

 = -- 



The magnitude of R may be found from the equa- 

 tion 



R + P sin. ft = W cos. a ; 



sin. a sin. B 

 or R = W cos. a P sin. ft = Wcos. a W - 



-w( 



cos. a cos. ft sin. a sin . ft\ W cos. (a 



cos. ft 



) 



cos. ft 



When P acts along the inclined plane in the direction 

 D B, ft = 0, and 



W _ 1 A B length of the plane , 



P sin. a B C height of the plane ' 



A C 

 also, R = W cos. a = W . ,, 



W X 



base of the plane 

 length of the plaue 



Correctly speaking, the angles, (fee., of (2) do not pro- 





perly belong to the inclined plane of (1), -unless the 

 centre of gravity of the body D touches the plane A B, 

 but to an imaginary plane passing through the point A 

 and the centre of gravity of D. 



THE WEDGE. The sixth mechanical power is tho 

 wedge, which may be regarded f lg _ 134. 



as two equal and similar inclined 

 planes A B C, B D C (Fig. 134), 

 with their bases fixed together. 

 It is used for cleaving substances, 

 in which case the edge C is in- 

 troduced into a cleft, and the 

 surface A D struck by a hammer 

 or mallet, so as to cause the 

 wedge to enlarge the cleft and 

 split the substance. 



Considered as a statical ma- 

 chine, the power is a weight ap- 

 plied to A D sufficient to balance 

 tho pressures exerted by the 

 substance into which the wedge 

 is thrust on the sides A C and D C of the wedge, con- 

 sidering these surfaces as perfectly smooth. 



The consideration of this machine is now omitted in 

 many treatises, since " in the theory of tho wedge, there 

 are introduced so many conditions, which are perfectly 

 inapplicable in practice, so many gratuitous assumptions 

 and suppositions so inconsistent with practical truth, 

 that the whole doctrine has little or no value. Nothing 

 can more plainly demonstrate tho hmtility of the theory 

 of the wedge, than that, in this theory, the power is sup- 

 posed to be a pressure exerted on the back of the wedge, 

 which is supposed to be capable of balancing the effect 

 of the resistance in producing the recoil of the wedge. 

 In all cases where the wedge is practically used, the 

 friction of its faces with the resisting substance is suffi- 

 cient to prevent the recoil ; so that, strictly speaking, 

 no force whatever is necessary to sustain the machine in 

 equilibrium ; and to move it, pressure is never resorted 

 to inasmuch as the slightest percussion is far more 

 effective. The only general theoretical principle re- 

 specting the wedge, which obtains always in practice is, 

 that its power is increased by diminishing tho an<*!o 

 CD."* 



THE SCREW. The seventh mechanical power is 

 the screw, and may be regarded as a modification of tho 

 inclined plane. If we take a triangular piece of paper 

 ABC (Fig. 135), the angle at C being a right angle, in 



Fig. 135 



A C take a point A', and through A' draw A' B' parallel 

 to A B. Then let the portion A A' B' B be blackened, 

 and wrap the triangle ABC round a cylinder D E. Let 

 now a g/oove be cut perpendicular to the surface of the 

 cylinder, and following the direction of tho dark spiral 

 band, formed by the triangle ABC. The projecting 

 portion between tho groove is called the thread of the 

 screw, and is evidently an inclined plaue passing round 

 the cylinder. The cylinder in which the groove is cut is 

 called the screw. A hollow cylinder is now cut through 

 i beam A B, and a spiral cavity, cut in the surface of 

 the hollow cylinder, corresponding in magnitude to tho 

 thread of the screw, so that ono may bo regarded as the 

 cast or mould of the other. This is called the nut of the 

 screw. 



" Mechanics;" Society for the Diffusion of Useful Knowledge. 



