BETWEEN SUEEACES MOVING AT LOW SPEEDS. 
517 
^2=0‘1188 foot per second. If a be the angular velocity of the disk, the angular 
acceleration = where R is the radius, or 09857 foot. The moment, M', of 
the couple due to friction, measured in absolute kinetic units, is I where I is the 
moment of inertia of the disk about its axis. Since the mass of the disk is 86-2 lbs., 
and its radius of gyration 0*697 foot, I=4T9. Hence, in the above example, M , =5 , 05. 
To reduce this to M, the moment of the couple due to friction where the force is 
expressed in terms of the gravitation unit, we must divide by 32-2 ; hence M=0T57. 
The value of this couple having been obtained, the coefficient of friction, expressing 
the ratio of the force due to friction to the normal pressure, remains to be found. 
During the revolution of the disk the axle presses against one side and the bottom of 
each bearing in the manner shown in fig. 7. If P„ be the pressure against the bottom, 
P A the pressure against the side, and W the weight of the disk, we have 
and 
Hence 
P*=I*P, 
P,=W— ^P*=W— pT,. 
P„ 
W 
1+/* 2 
1 P/x 2 
The couple due to friction (M) is (bP v r-\-[bP h r, where r is the radius of the axle. 
We have therefore a quadratic equation for determining 
^(Wr-M)+^Wr=M; 
or, to put it in a form better suited for arithmetical work, 
9 , W r Wr 
-TPWr-M — Wr-M"" 1 ' 
This equation has only one positive root. 
It has here been assumed that p is the same for both the places where sliding occurs. 
Even in cases where this assumption is not quite warrantable (as when the bearings are made 
of wood, and the motion is at the bottom in a plane perpendicular to the fibres, and at the 
side in the direction of the fibres), the amount of error so introduced into the determi- 
nation of fb will be exceedingly small ; for, since P„ is much greater than P A , the value of p, 
as above determined, is very approximately that corresponding to the bottom surface. 
Substituting for W and r their numerical values, viz. W=86 - 2 lbs. and r =- 004135 foot, 
we have 
, 0-3564 03564 
P 0-3564 — M 0-3564— M 
In the example cited M= 0-157 ; whence 
ja,=0"366. 
4 D 
MDCCCLXXVII. 
