MECHANICS. 



21 



plane WA would just put the weight in 

 motion. But we desire to know the 

 power which acting at any given angle 

 with "\VA would just move the weight. 

 Draw the line "\VB" in the direction of 

 the required power, and terminated in 

 AM ; the number of inches in ~\VB" will 

 be the number of pounds which, acting 

 in the direction TVB'', will just move the 

 weight. Again, it may be required to 

 assign the "direction in which a given 

 power must act in order just to move 

 the weight. To determine this let a 

 line of as many inches, as there are 

 pounds in the required power be inflected 

 fi-om W on the line AM. IHVB' be 

 this line, \VB' will be the required di- 

 rection. 



To determine the best angle of 

 draught, is then only to assign the least 

 line which can be drawn from the point 

 W on the line A M, which is, as is well 

 known, a perpendicular to it. Let 

 W B A be a right angle, and the angle 

 B ~\Y A is, therefore, the best angle of 



draught. The right-angled triangles 

 W B^A and B C A are similar, (Euc. 

 VI. prop. 8,) and, therefore, the angle 

 B W A is equal to the angle C B A. 

 But this last is equal to the angle to 

 which the plane should be elevated, in 

 order that the body should just move 

 down it. 



(26.) [We may obtain this result ana- 

 lytically thus. "Let x be the angle of 

 draught, P the drawing force, and X 

 the elevation at which the body just 

 moves down the plane. The elements 

 into which P is resolved are P cos. x, 

 and P sin. x. The pressure on the plane 

 is consequently W P sin. x, and the 

 corresponding friction 



(W - P sin. x tan X.) 

 This is balanced by P cos. x; therefore, 

 we have 



(AY - P sin. x) tan. X = P cos. x. : 

 The question then is to determine the 

 value of x, which renders P a maximum. 

 Differentiating considering P and x as 

 variables, we have 



P cos. x tan. X dx sin. x tan. X d P = cos. x d P P sin. x dx. 



Let d P = 0, and omit d x, and we 

 obtain 



P cos. x tan. X = P sin x. 

 .*. tan. X = tan. x ,\ x = X, 

 which is the conclusion obtained geome- 

 trically above.] 



(27.) Hence, if a body be required to 

 be drawn upon a plane subject to fric- 

 tion, the best direction for the traces is 

 to be inclined to the plane at that angle, 

 at which the plane itself should be in- 

 clined to the horizon, in order to make 

 the body move down it without any 

 drawing force. 



In the construction already instituted, 

 the angle of draught corresponding to 

 the direction W M is 90. In this case 

 the whole drawing force is spent in di- 

 minishing the pressure on the plane, 

 the element in the direction of the plane 

 gradually diminishing as the angle of 

 draught increases, and at length alto- 

 gether disappearing. The line "NY M 

 ought then to represent as it does the 

 weight of the body, the pressure being 

 in this case absolutely destroyed. 



(28.) In the preceding investigation 

 of the best angle of draught, we have 

 supposed that the plane upon which the 

 load is drawn is horizontal. If, how- 

 ever, it be not so but inclined, the pro- 

 cess of investigation will be somewhat 

 modified, but the final result will be the 

 same, the best angle of the draught 

 being in all cases equal to that elevation 



of the plane, at which the body would 

 just move down without any drawing 

 force. 



Let F I (/-.9.) be the inclined plane on 

 which the body is placed, and let its length 

 F I, expressed in inches, represent the 

 weight of the body, expressed in pounds. 



Fig. 9. 



Hence, its base F K will represent the 

 pressure on the plane, and its height 

 K I the force down the plane. (Second 

 Treatise). Draw W M perpendicular 

 to F I and equal to F K, and draw 

 M A, making the angle W M A equal to 

 the angle of elevation, which would just 



