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FRICTION AND LUBRICATION 261 
horizontal force P is applied to A, as shown in Fig. 392, and suppose 
that, on account of the friction between A and the plane, A remains at 
rest. P and W have a resultant S which makes 
an angle 8 with the normal to the plane and &® RV 
tan B=P/W, also S= ,/P?+W*=W/cos B. To B 
balance the force S there must be an equal and 
_ opposite force R exerted by the plane on A. Ifthe W Ww 
force P be increased and A still remains at rest, R 
will increase, and so will the angle 8. When P is Fic. 391. Fia. 892. 
increased until A begins to move, then P/W =p, by the definition of p, 
and the angle f will have its maximum value ¢, where tan¢=p. The 
angle ¢ is the angle which R makes with the normal to the plane when 
sliding begins, and is called the friction angle, the limiting angle of 
resistance or the limiting angle of reaction. 
If the plane be tilted up through an angle # and A remains at rest 
on the plane (Fig. 393), R, the reaction of the plane on A, must balance 
W, and must therefore make an angle with the normal to the 
plane equal to 8. The normal pressure of A on the plane 
is W cos f, and P, the component of W parallel to the 
‘el is W sin f. If the angle B be increased until A 
gins to slide down the plane, P will then be equal to 
pW cos B = W sin Pf, hence » = tan B=tan 4, and 4, 
which has been called the friction angle, is also the maximum =, gg 
inclination which the plane can have consistent with the O43) 
body A remaining at rest, or it is the minimum inclination which the 
plane can have consistent with the body sliding down the plane by the 
force of gravity. This inclination of the plane is called : 
the angle of repose, and it is the same as the friction 
e. 
Next let A be beginning to slide on a horizontal 
plane, the force P being inclined at an angle 6 to 
the horizontal (Fig. 394). The forces P, W, and R 
are in equilibrium, and R must be inclined to the 
normal to the plane at an angle ¢. From the triangle of forces, 
a sin } _ sing 
W sin (90+0-¢) cos (0-¢) 
’ 
Hence for given values of W and 4, P will be least when cos (@— ¢) is 
greatest, that is, when 9=¢; the direction of P will then be perpen- 
dicular to that of R. . : 
Consider next the case where a body of weight W is pulled up 
a plane which is inclined at an angle « 
to the horizontal by a force P acting 
parallel to the plane (Fig. 395), the motion 
being uniform. The forces which balance 
one another are P, W, and R, the latter 
force making an angle ¢ with the nor- 
mal to the plane. From the triangle of forces 
P_ sin (a+) _ sin (2+) _ sin acos $+cos asin ¢ 
W (sin90-¢) cosp cos p 
Fia. 394. 
=sin a+p cos a, 
