FRICTION AND LUBRICATION 277 
- resistance to rotation at the thrust bearing of the shaft, but in that case 
_ the resistance, reduced to the surface of the shaft, may be either greater 
or less than »Q, depending on the effective radius of the collar or pivot 
of the thrust bearing used. 
Consider now the work done in the given time when two guides are 
used, as in Fig. 421, each guide carrying half the load W. Neglecting 
the work done at the thrust bearings of the guides, the work done is 
2P-CL=2x 4uW-CL=pW-CL=U. If V isthe surface velocity of the 
rotating guides, and v the velocity of the sliding piece, 
2 
as RTL of and U =pW- oN. 
C 
The force T is equal to »W - se pW ce Vie e If the guides are at 
rest, V=0, and T=pW. 
From the foregoing, it is seen that the work done with rotating guides 
is greater than the work done with ordinary sliding guides in the ratio of 
,/V? +v? to v, and therefore the rotating guides would not be introduced 
to economise power. It would be absurd, for example, to use rotating 
guides in a planing machine. Rotating guides are useful in certain 
recording instruments, where a pen or pencil has to be guided in a 
straight line and moved by a small force. 
The same principle is also applied when it is required to reduce: 
the sliding friction of a piston or plunger in the direction of the 
axis, by giving the piston or plunger a simultaneous rotary motion. 
Kinematically, the mechanism in this case is the same as that discussed 
above. ; 
243. Friction of a Band on a Pulley.—Let a band ABCD (Fig. 
423) passing over a pulley have a tension T, 
in the part AB and a tension T, in the part 4 T+dT 
CD, and let the band be just on * the point of Tes 5 
slipping on the pulley in the direction from C to 
B. T, will be greater than T, on account of the 
friction between the band and ‘the pulley. Let 0 
be the angle subtended by the are of contact BC 
at the centre of the pulley. Consider an in- 
definitely small portion bc of BC subtending an 
angle do. at the centre of the pulley. Let T be 
the tension in the band at c, and T+dT the 
tension at b. Let S be the resultant of the pres- Fig. 423. 
sure of the pulley on the portion be of the band, and let p be the 
coefficient of friction between the band and the pulley. Then 
iT 
T+dT-T=dT=pS, but S=d6, therefore dP =pTd0, and Fy =pd0. 
Ty 0 
Integrating i =p 0, therefore log, 7 =p0, or rm otf. 
; Ty A 
In the above equations, @ is in circular measure, and the logarithm 
