MECHANICS. 



13 



intermediate place between the other 

 two, being less than the friction of 

 sliding, and greater than the friction of 

 rolling. 



To explain this friction, and the ex- 

 periments by which its properties may 

 be determined, let us suppose a solid 

 cylindrical axis, AB, (fig. 4,) inserted in 

 an hollow cylinder, of a diameter, C B, 

 somewhat greater than AB, so as to 

 permit the hollow cylinder, B C, to turn 

 round it, A B. Let the cylinders be 



Fig. 4. 



placed with their axes horizontal, and 

 let the hollow cylinder be the centre or 

 box of a wheel, D E. Let an extremely 

 flexible string be passed over the edge 

 of this wheel, in a grove formed to re- 

 ceive it, and let scales, G H, be append- 

 ed to its extremities. In consequence 

 of the form of the axle and hollow cy- 

 linder, and the manner in which the 

 weight of the wheel acts, the points of 

 contact of the axle and the cylinder will 

 be in a straight line, formed by the in- 

 tersection of a vertical plane passing 

 through the axis of the cylinder, with 

 the surface of the cylinder. In fact, if 

 from the point of contact, B, a line be 

 conceived to be drawn perpendicular to 

 the plane of the paper, along the inner 

 surface of the cylinder, the axle and the 

 cylinder will touch in that line, and in 

 no other points. It appears, therefore, 

 that if the hollow cylinder be supposed 

 to revolve round the axle, as happens 

 in a carriage wheel, every part of the 

 surface of the hollow cylinder is suc- 

 cessively exposed to the effect of fric- 

 tion ; while no part of the axle suffers 

 this effect, except the side which passes 

 through the point, B, of its section. If, 



on the contrary,~as sometimes happens, 

 the axle revolve within the cylinder, the 

 opposite effects are produced. The 

 entire surface of the axle is successively 

 exposed to the effects of friction, while 

 these effects are confined to one line 

 upon the surface of the hollow cylinder. 



By loading the dishes G H with any 

 equal weights, the axle may be submit- 

 ted to any proposed pressure. If, when 

 they are equally loaded, some fine sand 

 be poured into one of the dishes until its 

 weight just gives motion to the wheel, 

 the weight of the sand will be sufficient 

 to determine the quantity of friction. 



The preponderating weight is not, 

 however, in this case, the immediate 

 measure of the friction. It is to be con- 

 sidered that the wheel is turned round 

 its centre, I ; that the friction which 

 resists this motion acts at B, and there- 

 fore with the leverage B I ; while the 

 preponderating weight which overcomes 

 the friction acts with the leverage E I. 

 Let the friction be F, and the prepon- 

 derating weight be W ; then by the 

 established properties of the lever we 

 have 



F: w :: El :BI 



.-. F = w5JL 

 BI ; 



that is, the friction is equal to the addi- 

 tional weight which produces the mo- 

 tion, multiplied by the radius of the- 

 wheel, and divided by the radius of the 

 hollow cylinder which plays upon the 

 axle. 7 



Thus, it appears that the friction is 

 greater than the preponderating weight 

 in the proportion of the radius of the 

 wheel to the radius of the cylinder. 



As, in the experiments to determine 

 the friction of rolling, so here also each 

 experiment should be tried in both 

 dishes, and the mean of the results 

 taken. 



To determine whether the friction be 

 an uniformly retarding force, a weight 

 must be placed in one of the dishes 

 greater than that which is necessary to 

 overcome the friction. This will cause 

 the dish to descend with an accelerated 

 motion, and by placing a graduated 

 vertical scale near it, the rate of its ac- 

 celeration may be ascertained. If it be 

 found that the spaces through which it 

 descends, in one, two, or three seconds, 

 &c. are as the numbers 1, 4, 9, &c. ; in 

 other words, if the spaces be as the 

 squares of the times, the motion is uni- 

 formly accelerated. Hence it may be in. 



