PROP. XI. In the single moveable pulley where the strings are parallel, there is an equilibrium when the Power is to the Weight as 1 to 2. Let PA, which touches the pulley at A, be the line in which the Power (P) acts. Join A R with C the center of the pulley. Produce AC to B. Draw BR touching the circle; it is perpendicular to AB and parallel to AP; this therefore will be the other part B of the string. Let W, the weight, act at C by means of the block, in the line CW. The whole being at rest, the string RBAP must be equally stretched throughout, or motion would ensue. The force upwards therefore exerted by BR is P, for it is the same as that which keeps AP stretched. Now considering ACB as a straight lever kept at rest round C by forces P and P acting perpendicularly at A and B, the pressure upwards on C will be 2P (Prop. 11); wherefore the weight (W) which acts downwards at C and is supported by this pressure, is = 2P; or P: W:: 1 : 2. Wherefore, "In the single moveable pulley, &c." Q. E. D. [COR. The pressure produced at C by the forces P and P acting at A and B is perpendicular to AB, or parallel to AP and BR; wherefore the line of action of the Weight is perpendicular to AB, or parallel to AP and BR. If therefore "the Weight" be a heavy body hanging freely from C and acted on by the force of gravity, since in a case of equilibrium the strings are parallel to the line in which the weight acts, AP and BR are vertical.] 33. THE FIRST SYSTEM OF PULLEYS ;—where the same string passes round any number of pulleys. PROP. XII. In a system in which the same string passes round any number of pulleys, and the parts of it between the pulleys are parallel, there is an equilibrium when Power (P): Weight (W): 1 number of number of strings at the lower block. [In this Proposition, as in Prop. xI., the Power and the Weight are supposed to act in a direction parallel to that of the strings.] Since there is equilibrium, the portions of the string that lie on either side of any pulley must be pulled by equal forces; and since the outside portion, to which the power is applied, is acted on by a force P, each of the strings is therefore pulled by a force P. Wherefore the part of the weight sustained at A is (by Prop. x1) 2P, and the same is true with respect to the pulley B, and so on for every other pulley at the lower block. If therefore there be n pulleys at the lower block, W, the whole weight supported, will be n x 2 P; and we have, M Yw VP or P W :: 1 : 2n :: 1 : number of strings at the lower block. Wherefore," In a system, &c." Q. E. D. PULLEYS;-in 34. THE SECOND SYSTEM OF which each pulley hangs by a separate string. PROP. XIII. In a system where each pulley hangs by a separate string and the strings are parallel, there is an equilibrium when P: W :: 12 raised to that power whose index is the number of moveable pulleys. [The weight W is supposed to act in a line parallel to the strings.] In the figure, C being a moveable pulley with parallel strings, there is equilibrium if the F G II pressure downwards at C2 P. Wherefore, if W," the Weight," be = 23 P, there is equilibrium on the system here represented. And in the same manner, if n be the number of moveable pulleys, it would appear by the same mode of reasoning that there will be equilibrium when equal right-angled triangles, 35. DEF. If a prism whose ends are similar and (Cand cbeing the right angles), be placed so that either of the c sides of the prism adjacent to the right angles (as ACca) is horizontal, the slant side ABba is called an Inclined Plane. When the Inclined Plane is used as a Mechanical Power, both the Power and the Weight act in a plane parallel to either end of the prism, and the plane is represented by a triangle such as ABC, of which the side AB is called the Length of the plane, AC its Base, and CB its Height. And the plane, being supposed to be perfectly rigid and inflexible, will therefore be capable of counteracting and entirely destroying any force that acts upon it in a line perpendicular to it. PROP. XIV. The weight (W) being on an Inclined Plane and the force (P) acting parallel to the plane, there is an equilibrium when P: W the height of the plane its length. Let AB be the length of the plane; AC its horizontal base; and BC, per B F D pendicular to AC, its height. Let the Weight (W) act vertically at D, and the Power (P) act in the line DF which is parallel to the A plane; Then if P: W:: BC: AC, there is equilibrium. E Draw DE at right angles to AB, meeting the base in E; and EF vertical, or at right angles to AE. Then in the triangles EFD and ABC, angle DFE = angle ABC, · FE, BC are parallel, and AFB cuts them; and angle FDE = right angle = angle BCA, therefore angle DEF = angle CAB, and the triangles are equiangular. Now, by hypothesis, P: W: BC: AB :: DF: FE, by equiangular triangles. Now the same effect that is produced on D by the two forces that are represented in magnitude and in their lines of action by DF and FE, will be produced by their resultant. But their resultant (DE), being perpendicular to the plane AB, will be entirely counteracted and destroyed by that plane, and no motion would be communicated to D. The body at D, therefore, will remain at rest when acted on by the two forces P and W which bear the ratio to each other of BC: AB. Wherefore, "The Weight, &c." Q. E. D. |