The traces, plane, and whole earth.

these bodies are, in order,

P, R, and W.

The actions of

Of these actions or forces R has no effect on the

R

B

Fig. 72.

motion, being at right angles to it (compare page 76),

... P's effect=W's effect.

But P's effect is due to the whole of P, while W's is only due to the part of W acting in the line of motion, i.e. W cos EDA,

.. P=W cos EDA=W sin CAB,

which verifies the result before obtained.

Here, in all probability, the student will think that a great deal of time has been spent in arriving at a result which could have been obtained in one, or at the most two lines by "resolving along AC."

But it is just by failing to put the questions we have considered that very many errors are made in the treatment of this subject. No doubt in the present simple case a correct result would be arrived at; but, in the most complicated questions possible, the student will, by putting these questions in order, necessarily arrive at a correct result, allowing of course for oversights or errors of calculation.

It is instructive to examine two different ways in which we might have answered question (1).

Ist. We might answer-The waggon and traces. This is equally a body whose motion we can consider.

Then the answer to (2) becomes-The collar, plane, and earth; or, to (1), The waggon and all the harness; then to (2), The horse, plane, and earth.

It is perfectly immaterial which of the three methods of answering the questions we adopt, but we must take

one of the sets given complete. It would for instance not be correct to say in answer to (1) The waggon; and then to (2) The horse, because the horse does not touch the waggon, although he is of course in each case the natural source of the effort.

The student must clearly understand that it is only by thorough examination of the details of a question that we can arrive at a full understanding of it. And although such examination may in the present simple example appear even trivial, there are many cases in which neglect of such detailed examination leads to the most serious errors. We shall not be able from want of space to give this detailed process in every case, and shall use the ordinary expression "resolving the forces." But in every case the process will have been mentally gone through, which is the chief point; although, at any rate during the first study of the subject, the student will find it advantageous to actually write the process in full for each question.

There is one quantity which we have not found the value of because we could obtain our chief result without it, viz. the value of R. This is sometimes required, and to obtain "resolve in the direction of R" (see preceding).

Then

R=W cos CAB.

Case II.-Next let the effort be horizontal instead

of along the plane. This is not a practical case in an

ordinary inclined plane, but we shall use it for a particular case in the next chapter.

We have then P, W, and R-the effort, resistance, and reaction of the plane.

Then

and the inverse is the velocity ratio, for

Velocity of P in its own line of action = V cos CAB,

W

=V sin CAB,

V sin CAB

.. Velocity ratio=

V cos CAB

=tan CAB,

I

=

force ratio

To obtain R we resolve vertically, which gives

R cos CAB=W,

.. R=W sec CAB,

so that R is in this case greater than in Case I., the reason being that P helps to pull the slider or carriage against the plane.

The Wheel and Axle.-In this machine we utilise the turning pair to magnify an effort, which effect we produced in the inclined plane by the use of a sliding pair.

The motion has been already fully investigated in the preceding chapters, so we need little further description.

И

P

W

Fig. 74.

In Fig. 74 A is the wheel, B the axle, these forming one solid piece, the fixed frame, containing the bearings, forming the other.

The effort P acts on the

end of a rope, coiled round, and fastened to a point in

the wheel.

The resistance W acts on a rope similarly connected to the axle.

If r12. be the radii of the wheel and axle respectively, the condition for balanced forces is (page 65),

Pri Wr2,

For the velocity ratio take one revolution, then

Lift of W = length of rope coiled on axle,

Fall of P= length of rope uncoiled from wheel,

If the axis be horizontal we have the common windlass (Fig. 75), if vertical the capstan (Fig. 76). In the

W
Fig. 75.

Fig. 76.

first the effort is applied to a handle, which takes the place of the wheel, but comparing with chap. iii., page 64, we see that the shape of the piece does not affect the motion.

In the capstan the effort or efforts is applied to bars radiating from the head, as shown in the plan; the rope

lifting the weight passes away horizontally, and its direction is changed to vertical or any other required by passing it over guiding pulleys.

These two cases supply fresh examples of the application of a single force and of a couple. The windlass requires stout bearings, but the capstan being moved by forces applied in pairs at the ends of opposite bars, i.e. by couples, can do without any bearing at all at the top (compare page 66).

The Screw.-To use this, the third simple pair, as a machine, we apply a moment to effect the turning motion, either to the screw or nut, the other piece being prevented from turning; and we apply the resistance to resist the sliding motion of one piece, the other being prevented from sliding, i.e. the effort causes the relative turning and the resistance resists the relative sliding.

The double nature of screw motion makes this case a little more complicated than the preceding ones. In the inclined plane we generally regard the plane as a fixed body, since it is connected to the earth in nearly all practical cases. Also in the wheel and axle the bearings are considered as fixed; in the case of the windlass generally to the earth, and in the case of the capstan to the earth or to a ship; in either case they are fixed relatively to the observer.

But in the screw motion not only may either be actually fixed relatively to the observer, but one may be fixed against turning yet free to slide, while the other is fixed against sliding while free to turn, or vice versâ; the turning and sliding here mentioned being relative to the observer.

None of these considerations, however, will affect the relative motion, and hence we can at once apply the Principle of Work to give us the relation between the effort and resistance, and then examine some practical cases showing the different ways in which the pair is used.