278 APPLIED MECHANICS 
is the Napierian or hyperbolic logarithm. Using common logarithms, 
log pis 043430, and if m is the measure of the angle 6 in degrees, 
2 
T 
then log m= 0-00758pn. 
If the band lies in a V groove on the pulley, as shown in Fig. 424, 
this has the effect of increasing the resistance to slipping, because slipping 
must now take place on two surfaces (the sides 
of the groove), upon each of which the normal S * 
pressure is greater than half the normal pres- 20 
sure on a flat pulley. Considering the element 
be (Fig. 423), the resistance to slipping in the R 
V groove is 2uR=pS cosec a, where 2a is the \a R 
angle of the groove, but for a flat band the 
resistance to slipping of this element is pS. BIG 42) 
Hence the equations given above for a flat band will-apply to a band in 
a V groove if p be altered to p,, where p, =p cosec a. 
Exercises XVI. 
1. Prove the formule given under Figs. 425, 426, 427, and 428, the motion 
of the body of weight W being uniform and up the plane. 
&, P R gs, 
P 
a “ad a sei 
Fia. 425. Fi4g. 426. Fig. 427. Fig. 428. 
P sin(a+ tg PH gi i 
vw ie ze eee. Waco aay WT ones 
2. Referring to Fig. 427 for given values of W, a, and ¢, what is the value of 
6 when P is least ? 
3. Prove the formulx given under Figs. 429, 430, 431, and 432, the motion of 
the body of weight W being uniform and down the plane. 
Fig. 429. Fig. 430. Fia. 4381. Fig. 432. 
P_sin(¢-a) Pp. - re P _sin(¢-a) P_sin(¢-4) _ 
W  cos¢d ~ qo (pa). W cos(¢- 86) W cos(¢+0) 
4. For the key or cotter shown in Fig. 433, prove that the force P required to 
drive the key in is Q{tan(a+¢)+tan¢}, and that the force 
required to drive the key out is Q{tan (¢— a)+tan ¢}. 
5. What is the greatest taper which a key may have con- 
sistent with the friction holding the key in position? Take 
#.=0°07,and express the taper in the form 1 in a, where x isa 
length. 
