302 



PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 



13. The Modulus of the Pulley. 



Let Pj and Pg be taken to represent the moving and working (or the preponderating 

 and yielding) tensions upon the two parts of the cord passing over a pulley ; let 

 W represent its weight, a its radius measuring to the centre of the cord, ^ the radius 

 of its axis, and <p the limiting angle of resistance between the axis and its bearings. 

 Then if the cord were without rigidity, we should have by equation (13.), observing 

 that flj = «2 = «, and substituting W for P3, and f sin <p for X, 



Mp 



^ = {l+^«»^^ 



} 



W sin (p. 



m 



Up 

 La2 



W sin ^, 



But by the experiments of Coulomb (as reduced by M. Poncelet)*, it appears that 

 the effect of the rigidity of the cord is the same as though it increased the tension 



Pg so as to become Pg ( 1 + —J + — , where E and D are certain constants given i 



terms of the diameter of the rope. Taking into account the effect of this rigidity 

 the relation between Pj and Pg becomes therefore 



P.={,+^^.„.}{p.(. + |) + ?} + 



whence by reduction we have 



p. = (> + l){' + ^-^}p. + 7{> + (^+^J^-4. • (''■) 



where L represents the chord of the arc embraced by the string, and M the quantity 

 a^ (cos iij^ + cos 122), *i3 and ^23 being the inclinations of the two parts of the string to 

 the vertical (section 10.). 



Let the accompanying figure be taken to represent the pulley with the 

 cord passing over it, and E P3 the direction of the weight of the pulley, ! 



supposed to act through the centre of its axis, then are the angles /^^ and 

 /2 3 represented by P^ E P3, and P2 F P3, or their supplements, according as 

 the pressures P^ and P2 respectively act downwards, as shown in the figure, 

 or upwards-^", so that if both these pressures act upwards, then the 

 cosines of both angles become negative, and the value of M is negative; 

 whilst if one only acts upwards, then one term only of the value of M 

 assumes a negative value. Let the inclination A I B of the two parts of 

 the string be represented by 2 /, then L = A B = 2 a cos /. Substituting 

 this value for L, and also its value a^ (cos i^^ + cos 122) for M, and omit- 

 ting terms which involve products of the exceedingly small quantities 



— , — and — sm (p, we have 



a ' a 



* See PoNCELET, M^canique Industrielle, 128. 



p D W p (cos <t 3 + cos i2_3) sin </> 

 2 "^ « ' 2 a cos / 



t See Note, Section 9. 



