302 
PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 
13. The Modulus of the P alley . 
Let and P 2 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 a l = « 2 = a, and substituting W for P 3 , and £ sin <p for X, 
p i = { 1 + f sin p 2 + • Wsin 
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 
P 2 so as to become P 2 ( 1 + ~g) + where E and D are certain constants given in 
terms of the diameter of the rope. Taking into account the effect of this rigidity, 
the relation between Pj and P 2 becomes therefore 
P i - { 1 + 1? Sin { P 2 0 + a ) + a } + L a* W Sin 
whence by reduction we have 
p i = ('+!){ 
, . L p . 
1 + — F sin (p 
+ + & + ET5^)? sin « > }’ 
(23.) 
where L represents the chord of the arc embraced by the string, and M the quantity 
a 2 (cos q 3 + cos / 23 ), q 3 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 P 3 the direction of the weight of the pulley, [ 
supposed to act through the centre of its axis, then are the angles q 3 and 
/., 3 represented by P x E P 3 , and P 2 F P 3 , or their supplements, according as 
the pressures P x and P 2 respectively act downwards , as shown in the figure, 
or upwards -f~ ; 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 i, then L = AB = 2a cos i. Substituting 
this value for L, and also its value a 2 (cos / 13 + cos t 23 ) for M, and omit- 
ting terms which involve products of the exceedingly small quantities 
D E , p . . 
— , — and — sin <z>, we have 
a ’ a a r ’ 
', = {>+? 
. 2p 
+ — cos / sin <p 
1 a r 
} p 2 + T + 
W p (cos j, g + cos i 23 ) sin cf> 
2 a cos i 
* See Poncklkt, M^canique Industrielle, 128. 
f See Note, Section 9. 
