136 



Prof. H. Hennessy. Problems in 



[Feb. 24, 



and from the form of F'x in these problems ~F"x and Fx are positive, 

 therefore d 2 V/dx 2 must be always a positive quantity ; whence the 

 value of x obtained in all such problems makes V a minimum. 



Small pulleys carrying cords are usually made solid and approxi- 

 mately cylindrical ; in a train of such pulleys the volumes of the 

 large and small cylinders may be denoted by 7rZ>R 2 and 7rbr 2 , where 

 b is the common thickness of each cylindrical disk ; the total volume 

 of the train will therefore be — 



V= mirb(W + r 2 ) = m7r&r 2 (E, 2 /r 2 + l), 



or using the preceding notation, 



V= !*£2S^(i +( *)=K<l±A 



LOgX LOgX 



l_dV _ 2a 3 2 ]oga-(l + a; 2 ) 

 K dx ~ a (log a) 2 



which gives log x — ^(l + x~ 2 ) ; 



this equation is approximately satisfied by making x = 1"895. Hence 

 the ratio of the radii may be practically set down as 19 to 10 for a 

 train of pulleys of minimum volume or least weight of material. 



In drums the surface carrying the band is broad, and this surface is 

 commonly supported by spokes which radiate from the axle, while 

 sometimes, as in pulleys, the drum consists of a disk with a broad 

 hoop. If the thickness of the hoop and its disk are equal, a problem 

 similar to the foregoing can be easily solved. The question is, in a 

 series of large and small drums if all the large are equal and also all 

 the small, required the ratio of their diameters so that the entire 

 train shall have the least volume for a given velocity ratio. Let t 

 be the uniform thickness of the disks and hoops of the drums, R and 

 r the radii of a small and a large disk, b the breadth of the hoops ; we 

 shall have for the total volume of the train 



V = m?r[2(R + r)tb + t(B? + 7*)'] t 



when t is so small compared to R, r, and b, that quantities multiplied 

 by t 2 , &c, may be omitted. 

 The above may be written 



which gives, by the usual process of making dV/dx = 0, 



2(x+I)b/r + x*+l 



log X 



2x(blr + x) 



