258. The mechanical powers now to be considered are often used for
other purposes beside those of raising great weights. For example: the
parts of a structure have to be forcibly drawn together, a powerful
compression has to be exerted, a mass of timber or other material
has to be riven asunder by splitting. For purposes of this kind the
inclined plane in its various forms, and the screw, are of the greatest
use. The screw also, in the form of the screw-jack, is sometimes used
in raising weights. It is principally convenient when the weight is
enormously great, and the distance through which it has to be raised
comparatively small.
[Illustration: FIG. 41.]
259. We shall commence with the study of the inclined plane. The
apparatus used is shown in Fig. 41. A B is a plate of glass 4' long,
mounted on a frame and turning round a hinge at A; B D is a circular
arc, with its centre at A, by which the glass may be supported; D C
is a vertical rod, to which the pulley C is clamped. This pulley can
be moved up and down, to be accommodated to the position of A B; the
pulley is made of brass, and turns very freely. A little truck R is
adapted to run on the plane of glass. The truck is laden to weigh 1
lb., and this weight is unaltered throughout the experiments; the
wheels are very free, so that the truck runs with but little friction.
260. But the friction, though small, is appreciable, and it will be
necessary to measure the amount and then endeavour to counteract its
effect upon the motion. The silk cord attached to the truck is very
fine, and its weight is neglected. A series of weights is provided;
they are made from pieces of brass wire, and weigh 0·1 lb. and 0·01
lb.: these can easily be hooked into the loop on the cord at P. We
first make the plane A B horizontal, and bring down the pulley C so
that the cord shall be parallel to the plane; we find that a force must
be applied by the cord in order to draw the truck along the plane: this
force is of course the friction, and by a suitable weight at P the
friction may be said to be counterbalanced. But we cannot expect that
the friction will be the same when the plane is horizontal as when the
plane is inclined. We must therefore examine this question by a method
analogous to that used in Art. 207.
261. Let the plane be elevated until B E, the elevation of B above
A D, is 20"; let C be properly adjusted: it is found that when P is
O·45 lb. R is just pulled up; and on the other hand, when P is only
0·40 lb. the truck descends and raises P; and when P has any value
intermediate between these two, the truck remains in equilibrium.
Let us denote the force of gravity acting down the plane by R, and
it follows that R must be 0·425 lb., and the friction 0·025 lb. For
when P raises R, it must overcome friction as well as R; therefore
the power must be 0·025 + 0·425 = 0·45. On the other hand, when R
raises P, it must also overcome the friction 0·025, therefore P can
only be 0·425 - 0·025 = 0·40; and R is thus found to be a mean between
the greatest and least values of P consistent with equilibrium. If
the plane be raised so that the height B E is 33", the greatest and
least values of P are 0·66 and 0·71; therefore R is 0·685 friction
0·025, the same as before. Finally, making the height B E only 2", the
friction is found to be 0·020, which is not much less than the previous
determinations. These experiments show that we may consider this very
small friction to be practically constant at these inclinations. (Were
the friction large, other methods are necessary, see Art. 265.) As in
the experiments R is always _raised_ we shall give P the permanent
load of 0·025 lb., thus sufficiently counteracting friction, which we
may therefore dismiss from consideration. It is hardly necessary to
remark that, in afterwards recording the weights placed at P, this
counterpoise is not to be included.
262. We have now the means of studying the relation between the power
and the load in the frictionless inclined plane. The incline being
set at different elevations, we shall observe the force necessary to
draw up the constant load of 1 lb. Our course will be guided by first
making use of the principle of energy. Suppose B E to be 2'; when the
truck has been moved from the bottom of the plane to the top, it will
have been raised vertically through a height of 2', and two units of
energy must have been consumed. But the plane being 4' long, the force
which draws up the truck need only be 0·5 lb., for 0·5 lb. acting
over 4' produces two units of work. In general, if _l_ be the length
of the plane and _h_ its height, _R_ the load, and _P_ the power, the
number of units of energy necessary to raise the load is _R h_, and the
number of units expended in pulling it up the plane is _P l_: hence
_R h_ = _P l_, and consequently _P_ : _h_ :: _R_ : _l_; that is, the
power is to the height of the plane as the load is to its length. In
the present case _R_ = 1 lb., _l_ = 48"; therefore _P_ = 0·0208 _h_,
where _h_ is the height of the plane in inches, and _P_ the power in
pounds.
263. We compare the powers calculated by this formula with the actual
observed values: the result is given in Table XIII.