40 



MECHANICS. 



termined, whatever be the system of 

 pulleys. In fig. 72, the system of pul- 

 lies described in (83) is represented with 

 the ropes oblique. The tension of the 

 first rope, P A", (fig. 72.) is equal to 



CHAPTER IX. On the Inclined Plane. 



(91.) THE INCLINED PLANE is a ma- 

 chine formed, as the name imports, by 

 a plane surface, supposed to be perfectly 

 hard and inflexible, and which is always 

 inclined obliquely to the weight or the 

 resistance to be overcome. 



Let the line L H (fig. 73.) be hori- 

 zontal, and let L M be a perfectly hard 

 and smooth plane, forming, with the 

 horizontal plane, the angle M L H, 

 called the elevation of the plane. The 

 line L M is called its length, M H its 

 height, and L H its base. 



Fig. 72. 



the power. Let the tension of the se- 

 cond rope be t. Then by (88) we have 



t = 2 P Cos. A", 



A" being the angle under the parts of 

 the first rope. 



In like manner, if t' be the tension of 

 the rope A A', we have 



t' = 2 t Cos. i A', 

 and in the same way 



W = 2 t' Cos. i A. 



Multiplying these three equations to- 

 gether, we have 

 W t' t = P t' t Cos. $ A Cos. | "A 



Cos. i A". 



Omitting the common multipliers t' t, 

 we have 

 . W=8 P Cos. J A Cos. i A' Cos. \ A". 



It is easy to see how a similar inves- 

 tigation may be extended to any case 

 in which the ropes are oblique. 



(90.) Friction has always been a great 

 source of waste of power in pullies. 

 This, however, has been in a great de- 

 gree removed by an ingenious contri- 

 vance of Mr. Garnet, called friction- 

 rollers. They not only save expense 

 and labour, but also considerably di- 

 minish the wear of the machine. The 

 principle is this : between the axis on 

 which the wheel turns, and the con- 

 cave cylinder or box in which that axis 

 is placed, a hollow space is left to be 

 filled by solid equal rollers, nearly 

 touching each other. These are fur- 

 nished with axles, inserted in a cir- 

 cular ring at each end, by which their 

 relative distances are preserved, and 

 they are kept parallel by means of 

 wires fastened to the rings between the 

 rollers, and which are rivetted to them. 



Let A be a weight placed upon this 

 plane, and sustained by a power in 

 any direction, as AC. The body A 

 is kept at rest by three forces acting at 

 its centre of gravity : 1, the force of 

 gravity, W acting in the vertical di- 

 rection, AW; 2, the power P acting 

 in the direction A C ; and 3, the re- 

 sistance R of the plane acting in the 

 direction A B perpendicular to the 

 plane. Now, since the weight W, in 

 the direction A W, resists the forces P 

 and R, in the directions A C and A B, 

 it must be equal and opposite to the re- 

 sultant of these two forces. (Treatise 

 I. Chap. II.) Suppose A D drawn 

 directly upward, in the direction W A, 

 and from any point D draw D C and 

 D B, parallel to A B and A C respec- 

 tively, and it follows that the weight, 

 the power, and the resistance of the 

 plane will be proportional to the lines 

 A D, A C, and A B. This may easily 

 be verified by experiment. On a ver- 

 tical plane behind the power and 

 weight, draw the line A D vertical, 

 and from any point D in it draw D C 

 in a direction at right angles to the 

 plane. Upon measuring the lines A D 

 and A C, they will be found to have 



