42 



OF PRESSURE AXD EQUILIBRXUHr, 



of the c)'linilerj as the length of a coil to the 

 length of the axis. 



Let AB anil BC be tangents of the 

 small arc AC ; then if BC, and BA re- 

 present the force of tension, at A and C, 

 the diagonal of the parallelogram BD 

 will be the joint result ; but BD=2BE, 

 and by the properties of similar triangles 

 BE : BC : : EC : CF, BD : BC : : AC ; 

 CF. If the position of the thread be oblique, we shall find 

 by the composition of forces, supposing it uncoiled, and its 

 extremities retained in a line parallel to the axis, that its 

 tension is to a force acting in the direction of the axis, as 

 the oblique length of any portion to its height. Now this 

 tension produces on any small oblique portion of the cir- 

 cumference, a pressure equal to that which would be pro- 

 duced on the corresponding transverse portion by an equal 

 force acting transversely ; for the versed sine of the arc is 

 the same in both cases ; consequently the pressure on the 

 whole circumference is equal to that which would be pro- 

 duced by the same tension acting transversely. 



300. Theorem. The perpendicular pres- 

 sure of a weight resting on an inclined plane, 

 and retained in its situation by u resistance in 

 the direction of the plane, is to the weight, 

 as the horizontal length of the plane to its 

 oblique length, and to the resistance, as the 

 horizontal length to the height. 



The truth of the proposition is evident from the propor- 

 tioiu of the sides of the triangle conesponding to the direc- 

 tions of the forces. 



Scholium. Hence the proportion of the friction to the 

 weight may be determined by measuring the tangent of the 

 angle at which the weight begins to slide down the plane. 



301. Definition. A wedge is a solid 

 Included by two equal triangles joined by 

 thi'ee rectangles ; and we shall suppose the 

 surfaces to be perfectly smooth. 



302. Theorem. Three forces acting di- 

 rectly on the sides and base of a wedge will 

 be in equilibrium when each force ispropor- 



y tional to the side on which it acts ; provided 

 l!hat they be all applied at such parts, that 

 thek directions may meet in one point. 



For the triangle formed by three 

 lines perpendicular to the sides of 

 another triangle, is equiangular with 

 it, and if the forces act completely on 

 any point, it will remain in equilibrium 

 (2S4). •• 



303. Theorem. Supposing a moveable 

 inclined plane, orarectangular wedge to slide 

 without friction on a horizontal plane, it will 

 remain in equilibrium with a weight acting 

 vertically, when the horizontal force is to 

 the weight as the height to the horizontal 

 length. 



-% 



c 



The triangle ABC is similar to 

 ADB, and to BDC; and if AD 

 represent the weight, its perpen- 

 dicular pressure on the plane will 

 be AB (284), which will be held 

 in equilibrium by a force on the 



base, which is to it, as BD to AB (302) ; and this force will 

 be to the weight, as BD to AD, or as CD to BD. 



304. Definition. By rolling a thin and 

 flexible wedge round a cylinder, we form a 

 screw. 



305. Theorem. A force acting in the 

 direction of the circumference of a screw, 

 supposed to move freely round its axis, will 

 counterbalance a weight pressing vertically 

 on the screw, which is to it as the circum- 

 ference is to the height of one spire. 



For when the horizontal length of the wedge becomes 

 equal to the circumference of the circle, its height is the 

 height of the first spire of the screw, or the distance be- 

 tween any tvro spires or threads. 



Scholium. The cylinder may be either convex or con- 

 cave, making a cylindrical or a tubular screw, together 

 sometimes called a screw and a nut. The nut acts on th« 

 screw as a single point would do, only dividing the pres- 

 sure. In general the screw is applied in combination with 

 a lever. 



306. Theorem. If it be required to find. 

 the position of four equal beams capable of 

 supporting each other in equihbrium, two of 

 them fixed at the extremities of the base of 



