APPLIED MECHANICS. 



[DBUMS CO.XICAL rtnxrrs. 



makes one ; or that the angular velocity of the 

 mailer pulley U jut double that of the larger, because 

 it* circumference U half the length. If we take the 

 circumferences in any other ratio, we should find the 

 angular velocities in the inverse ratio. As the circum- 

 ferences of circles are exactly proportional to their 

 diameters, the angular velocities of pulleys connected 

 by straps are inversely as their diameters, or the diame- 

 ter of each pulley, multiplied by its speed, gives a like 

 product. In calculating the speeds due to given dimon- 

 fiont of pulleys, or the dimensions suited for given speeds, 

 we have therefore the following simple rules : 



Given the angular velocity of one of two pulleys (. e., 

 the number of revolutions per minute), and the diameters 

 of I- >th, to find the speed of the other. 



JI'H/C. Multiply the speed of the first by its diameter, 

 anil divide by the diameter of the other. 



Example. A pulley 30 inches in diameter, making 

 80 revolutions per minute, drives a pulley 24 inches in 

 diameter : required the speed of the latter. 



XT- " 120 revolutions per minute. 



Given the angular velocities of two pulleys and the 

 diameter of one, to find that of the other. 



llule. Multiply the speed of the first by its diameter, 

 and divide by the speed of the other. 



imple. A pulley 24 inches diameter, making 120 

 revolutions per minute, drives another at 80 revolutions 

 per minute : required the diameter of the latter. 



120x24 

 - gjj 36 inches. 



Pulleys are often combined with direct or crossed straps, 

 as may be required, in order to vary the velocity and 



direction of rotation. Thus the pulley A (Fig. 225), 24 

 in. lie* diameter, revolving at the rate of 10 revolutions 

 per minute, drives B, 12 inches diameter, at 20 revolu- 



tic, becau* *12 - 20. 



On the shaft of B is 



fixed a pulley C, 36 inches diameter, revolving at the same 

 spend with B, and driving D, 12 inches diameter, by a 



crossed strap at CO revolutions, because ~^ - CO. 



Again, on the shaft of D in another pulley E, 27 



uches diameter revolving with a speed of 60, and 



driving F, 9 inches diameter, at a speed of 180, because 



0>X 27 



- 180. 



In entimtintf the final speed produced by such a train 

 <* pulleys, it is nut necessary thus to calculate the velo- 



city of each, as we find the same result by multiplyim- 

 the initial speed of the driver A, by the product of the 

 diameters of all the drivers A, C, and E, and dividing by 

 the product of the diameters of those driven, B, D, and 

 F ; or 

 10 revoKx 24X30X27 



12 X 12 X 9 revolutions, the speed of F. 



Sometimes, when it is desirable to have the power of 

 varying the speed of machinery driven by a strap, the 

 pulleys are elongated and made conical, and the strap is 

 Fig. JJ6. 



shifted longitudinally to such a position as shall give the 

 speed required. Thus, if a a bo the driving-shaft, re- 

 volving 24 times per minute, and it be desirable that b b 

 shall be driven at any speed from 8 to 72 revolutions per 

 minute, each pulley being made conical with the larger 

 end 3 times the diameter of the smaller, the strap c may 

 be shifted from one end to the other to give the desired 

 variation of speed. Nor is the tightness of the str.tp 

 materially altered ; for as it is made to {Miss round a 

 larger circumferenceof the one, it passes round a smaller 

 circumference of the other, and so gains nearly as much 

 as it loses. Most of the machines that are driven by 

 straps do not require so nice an adjustment of velocity, 

 but act very well at various speeds within certain limits. 

 In such cases the conical pulleys are made with regular 

 steps, or consist each of a set of pulleys placed side by 

 side, having diameters increasing as much in the one as 

 they decrease in the other (Fig. 227). Thus, if the 



Fig. 227. Fig. 223. 



driving-shaft make 48 revolutions per minute, the driven 

 shaft may be made to revolve at the following different 

 speeds : 



