262 APPLIED MECHANICS ~ 
If b, h, and 7 be the base AC, height BC, and length AB of the plane — 
respectively, then sina=h/J and cosa=b/I, therefore woe b and 
- Pl=Wh+pWoe, which shows that the work done in drawing a body up 
an inclined plane is equal to the work done in lifting tt against gravity 
through a height equal to the height of the plane, plus the work done in 
drawing tt along the base of the plane against friction. This is a useful 
rule to remember. 
Here it may be pointed out that when a is comparatively small, as 
it generally is for most roads and railways, it is 
sufficiently accurate to assume that the base and 
length of the plane are equal. 
A case of the inclined plane which is import- 
ant in connection with the theory of the screw, 
is that in which the force, P is parallel to the 
base of the plane (Fig. 396). The triangle of forces shows that 
P=W tan (4+ ¢). 
230. Efficiency of the Inclined Plane.—The efficiency of any machine 
being the ratio of the useful work done to the total work, this must be 
- the same as the ratio of the effort when friction is neglected to the effort 
when friction is considered. Taking the case of the inclined plane shown 
in Fig. 395, where the effort P acts parallel to the plane, it has been 
shown that p. wen cr®) when friction is considered. If =0, 
P=W sin a, which is the value of the effort when friction is neglected. 
sin a cos 
sin (a+) : 
For the case shown in Fig. 396, where the effort is horizontal, the 
tan o. i 
tan (a+) 
231. Friction of Screws.—The connection between the inclined 
plane and the screw is shown clearly by 
Figs. 397 to 401. In Fig. 397 is shown 
a cylinder with one turn of a_ helix 
traced on its surface ; the dotted right 
angled triangle is the development of 
the portion of the surface of the 
cylinder which is below the helix. p 
being the pitch of the helix, a its in- Fic. 397. 
clination, and d the diameter of the cylinder, tan a= p/7d. . 
In Fig. 398 the inclined plane and the body sliding on it are two 
similar wedges which, when bent round a cylinder, as shown in Fig. 399, 
produce a form of screw and nut. 
The connection between the inclined plane and a square double 
threaded screw and nut is shown in Figs. 400 and 401. 
The force P in Figs. 398 to 401 is shown acting parallel to the base 
of the inclined plane\or perpendicular to the axis of the screw, and in the 
case of the screw, P acts at a distance from the axis equal to the mean — 
Fig. 396. 
Hence the efficiency in this case is —————* 
efficiency is 
