CHAPTER VII

SIMPLE MACHINES WITH FRICTION

THE results obtained in the last chapter will not hold in actual machines, since they neglect the influence of friction. Hence those results were too favourable; but at the same time they are of value as showing a limit which should be approximated to as closely as possible and as a rough guide to what we may actually expect. Also by neglecting the friction at first we are able more easily to grasp the problems to be solved than if we commenced by taking it into account; we thus follow the principle of introducing our difficulties singly if possible, but, at any rate, gradually.

The velocity ratios found are not affected by the question of friction, they being determined geometrically, not statically; but now we cannot proceed, as before, by the Principle of Work, because it takes the form of

Energy exerted work done + work wasted,

and the work wasted, depending on the pressures between the moving surfaces, will necessitate those pressures being found by the principles of Statics, which in the last chapter we only used for verification.

Also we now introduce a new conception, viz. :—

Efficiency. When we move a machine, or a pair, then actually there is always work wasted, so that the work done is less than the energy exerted. The ratio

of the work done to the energy exerted is called the Efficiency of the machine, or the pair.

We use whichever is the most convenient to calculate. Sometimes we need a term to represent the reciprocal of the efficiency, and this we call the Counter Efficiency.

the letter e being often used for the purpose here shown, of denoting the ratio of work wasted to work done.

We have

Efficiency=

work done energy exerted'

resistance x distance it moves

effort x distance it moves

= force ratio x velocity ratio,

so that we see that we cannot now determine one of these from the other without knowing the efficiency. Since the efficiency is necessarily less than unity, it follows that for a given velocity ratio the force ratio is less than it would be if there were no friction (page 96), so that the mechanical advantage is decreased by the friction.

[The term efficiency is sometimes used with very different meanings, and it is necessary to be very careful to find out, in

any given case, exactly what is meant by it. This remark applies generally, and not especially to our present subject. What we mean by efficiency has been already stated, and in that sense only will the term be used.]

We will now commence with the
Inclined Plane.

Comparing with Fig. 72, page 98, we have an extra force acting, because the action of the plane is now a double one, although there are still only the same number of bodies acting. We see then that in answering the questions on page 97 we must be careful to see that we take account of the total action of each body.

FR

A

Yw

Fig. 105.

Hence we must

The extra force is ƒR, where ƒ is the coefficient of friction (page 51). determine R by Statics. This gives, since ƒR and P have no effect in the direction of R, the effect of W in that direction equal to the effect of R, i.e.

W cos CAB=R,

as on page 99, the extra force ƒR not altering the equation,

Then

..ƒRƒW cos CAB.

Energy exerted = P. AC,

taking the movement from A to C.

And here we may introduce another caution, in addition to those of page 97, viz. that we must clearly define in our minds exactly the period during which we intend to apply the equation or principle of work. The period now taken is from the instant of starting from A to that of arriving at C.

and

Resuming, we have

Work done =W x CB,

Work wasted=ƒR × AC,

=fW x AC cos CAB,
=ƒ. W × AB.

And

... P. AC=W.BC +ƒW.AB.

The force ratio is now

W

AC
PBC+f. AB'

being, as we have already stated it must be, less than the reciprocal of the velocity ratio.

The foregoing work applies certainly to the case where W actually slides on the plane, but does it now apply to the pseudo sliding of a wheeled vehicle?

We have already stated that such resistance does, on a horizontal plane, practically follow the laws of sliding friction; but at the same time we know that actually the resistance arises from the axle friction, and also from the peculiar action between the rolling wheels and the ground, for an explanation of which we must refer to the larger treatise. It might seem then at first sight that the amount of resistance would be, not fR, but ƒW, because the carriage still rests on its axles as it does on a horizontal plane. But further consideration shows that the whole weight does not now rest on the axles, because the pull P tends to lift the carriage off them; and it does this exactly in the same proportion as it would relieve the plane of a part of the weight of a sliding piece, and hence the friction is reduced in the proportion of R to W, i.e. it is ƒR or ƒW cos CAB as for a slider.

In ordinary cases of gentle inclines it is not, however, practically necessary to consider this, because cos CAB approaches so nearly to unity. Thus on a slope of 1 vertical to 50 along the road

We have treated motion up the plane, but we have also motion down the plane to consider.

In this case P is down the plane and ƒR up. Also W is now an effort as well as P.

Energy exerted=P.AC+W.BC,
Work done=0,

Work wasted=ƒ. R. AC,

Then

=fW.AB, as before.

There is no work done in the sense

of useful or recoverable work, such as

YW

Fig. 106.

the lifting up of the weight, the energy exerted in which can be recovered by allowing it to fall again; so that the efficiency is zero. But we must not take this to mean that there is no useful effect, because we wish the carriage or train to go down the slope, and so the effect produced is what we required. We must, however, be consistent, and so we say all the work is waste work. The equation of work is then

and hence may be either positive, zero, or negative, as f. AB is > or <BC.

The meaning of this is, that, taking say a train— In the first case, to keep the motion uniform the engine must exert a pull.

In the second, the train will just run uniformly by itself.

And in the third, there must be a backward push; this would not, however, be applied as an actual push, but by applying the brakes, ƒ would be increased till f. AB became equal to BC.

And we have thus a means of determining, experi