it will not in all cases be so evident, and we may have a balance without such equality of effort and resistance in lbs. as we have here (see turning pair, page 64). To all such cases, however, the principle of work, in its simplest of all forms as just given, will apply. And we are not to look upon this statement as derived from other principles, but as in itself a first principle, which we can in any instance verify, but which must not be taken to require proof.

The principle of work is only a statement, for this case, of the conservation of energy, and we will now see how that doctrine reduces to its present simple form.

The case to which we say it applies is that of Balanced Forces. Now, if the forces are balanced, there is no excess left to produce changes in the body moved. The body moved is the agent by which the energy causes the doing of the work, and we can also then group all these cases under the heading of No change produced in the agent used. Using this definition, there is no effect produced by the energy exerted except the doing of work. Hence by conservation of

energy

Energy exerted = sum of all effects produced,

= work done only.

For example :-A railway truck weighs 2 tons, resistance 20 lbs. per ton. Find the energy exerted in keeping it moving at a constant speed through 50 yds. This is a case of balanced forces, and we have

Energy exerted = effort × distance,

=40 lbs. x 150 ft.,

6000 ft. lbs.

Had the question been set us to move the truck 50 yds. instead of "keeping it moving at a constant speed," we should not have a case of balanced forces, because the process would involve starting and stopping, during which processes, as we shall see hereafter, the forces would not be balanced.

But the larger statement of "no effect on the agent" would hold good, because the state of the truck would have been on the whole unaltered, and the principle of work in its simplest form applies. There is, on the whole, a balance, but not at each instant. This we shall not now inquire into, but only note it to show that the simple statement of the principle applies to such cases, i.e. motion from rest to rest again.

In the two cases of sliding we have so far considered, one body has, in each case, been the earth; but this is not at all necessary. For in any case whatever

Energy exerted, or work Resistance to relative motion x distance moved relatively.

done,

=

For example:-To plane a piece of material we may use a machine, in which the piece is fastened to a moving bed, the tool being still (Fig. 43); or one in which the tool moves, the work being still (Fig. 44).

If now R be the resistance offered by the metal to the point of the tool, and I be the length of the work, in lbs. and ft. respectively. Then evidently, in each case,

Work done in one cut = R x 7 ft. -lbs.,

quite irrespective of which moves. But there is this difference, that the energy is exerted in opposite manners in the two cases. In the first case the effort moves the work against the resistance of the tool. In the second the effort moves the tool against the resistance of the

work. Both require the same amount of energy, but it must be applied to a different part of the machine.

["The work" above means the piece of material, that being the usual workshop term.]

In addition to the examples we have already considered, the principal examples of sliding pairs, in which energy and work require calculation, are engine pistons and pump plungers.

With regard to the first, all we need notice is that it is immaterial whether the cylinder be still or whether it oscillate, the calculation of energy is unaltered.

In pumps the work to be done is the lifting of water, and we can simplify the calculation in such cases by the following methods :

W 2

Lifting of a Number of Weights.-Let W1,

etc., be the weights, and let them be initially at heights Y Y2, etc., above a given datum level.

Let now W1 be lifted a distance y1, W2 a distance 12, etc.

1

Their new heights are then Y1+Y1, Y2+2, etc., which we will call Y'1, Y'2, etc. Then

Work done = W11+W22+

=W1(Y'1 - Y1)+ W2(Y'2 − Y2) +..

In the case now of a shaft full of water, we need not

consider the separate lifts of the pump, but only the total lift of the C. G. of the whole body of fluid; or equally, in calculating the energy which the water stored in an elevated reservoir could exert by descending.

Motion in a Curve-Turning.—It is not necessary that the relative motion of two pieces be rectilinear sliding for the preceding calculations to apply, so long as the resistance is directly opposed to the motion. For consider such a motion as a piece along a curve AB, the paper representing the other

Fig. 45.

piece. Then, by dividing the curve ZIKË up into a large number of small A portions as ab, of length x, we have a number of rectilinear slidings each giving

Work done Rx,

and finally, for the whole,

Work done = Rx sum of x's,

= Rx curved length AB,

but the italics above must be carefully attended to. We will now consider the turning pair balanced. The turning pair consists of a shaft, turning in a bearing, and having arms to which efforts and resistances can be applied.

We must first determine the necessary conditions for balance. If there be only one arm, as in Fig. 46, and the effort and resistance be applied to it exactly opposite to each other, then evidently PR, i.e.

Fig. 46.

But if, as is generally the case, the resistance be not applied to the same arm, or if to the same arm not directly opposite, we must have some different condition. In Fig. 47 P and R are applied to arms of different lengths.

Now we can determine the condition of balance, either from Statics or by the principle of work. We I will select the latter method.

Let the crank move to the new position as shown,

B to B', C to C' (Fig. 48), the centre lines only being shown, then

This then is the condition of balance, or in words

Moment of effort = moment of resistance.

But this is the principle we know in Statics as the Principle of the Lever, and we have thus proved this principle by means of the Principle of Work. Or we may say, if we please, that we have verified the Principle of Work. We have in Fig. 47 drawn both P and R at right angles to the cranks on which they act. This was for simplicity, but we shall now see that there is no necessity for this to be the case.

For example, let the turning body, outside the bearing,