Turning moment of engine=W7.

[In comparing with the preceding take as the first body the piece of shaft inside the blocks, and consider the remaining part as an external body acting on it.]

There is one point in which the tail-rope brake has an advantage over the Prony; which is that any small discrepancies between (W-Q)r and M, caused by slight variations of M, or by inequality of lubrication altering F, and hence W - Q, are automatically adjusted. For suppose M increases slightly, becoming momentarily greater than (W-Q)r; then (Fig. 164) M carries the rope round to the left; but this decreases Q, and, therefore, W being constant, increases W-Q; and so the balance is restored. If on the other hand M fall off, W pulls the rope round to the right, increases Q and therefore decreases W-Q, restoring the balance once more. Next suppose the lubrication fall off a little, then the coefficient of friction increases; so that the wheel seizes the rope, carrying it slightly to the left; this decreases Q and therefore also W+Q, but the total amount of friction depends on W+Q the total load, so that the friction at once falls off and the rope slips back. Similarly for an increase of lubrication the balance restores itself. In the Prony brake these inequalities would keep the brake continually striking against one or other of the stops.

In actual practice then the brake is not still but oscillates continuously; but we must be careful to notice that this does not affect our results, because there is, on the whole, no velocity imparted to the brake (compare page 60), the small oscillations simply cancelling each other.

[Various means have been adopted for producing an automatic adjustment, but space prevents our examining them. If the

student should have occasion to examine such cases, let him be very careful to consider the action on the brake of every body that touches it. Examples of the utter confusion arising from not attending to this point may easily be found.]

Integrating Dynamometers. · - Great accuracy can be obtained by making the dynamometer register the force transmitted on a paper band which is moved by connection to the engine shaft, thus a curve is traced out, the area of which gives the work done. We may look upon it as a sort of continuous indicator diagram, the principle being the same. Such dynamometers are

called Integrating Dynamometers.

Brakes and Governors.--The absorption dynamometer can be used as a brake to absorb surplus energy, and in this case the problems arising are solved exactly as those already treated. If now we have an engine working at constant power, while the work required to be done varies, we could keep the speed constant by means of a brake, so adjusted as to absorb the surplus energy of the engine. But this would be a wasteful process, since the surplus energy would be wasted, hence for such a purpose we use, not a brake, but an instrument which cuts off partially the supply of energy, or of

C

AB is a spindle rotated by the engine, so that its angular velocity bears a constant ratio to that of the shaft. CC are heavy balls connected by rods DD to a point B of the spindle, then as the spindle revolves the balls fly out and pull up the piece G, which can slide on the spindle, the pull being applied by the

Fig. 167.

rods EE. The slider G is connected by linkwork, not shown, to the regulating valve; and the proportions are so arranged that, when the engine is running at the proper speed for the work it has to do, the regulating valve is just wide enough open to enable the engine to run against the average resistance. Suppose now the resistance rise above this mean value, then the speed falls off, consequently the balls fall in towards the spindle, pushing down the slider, and the linkwork is so arranged that this opens the regulator, admitting more steam, and so enabling the engine to overcome the increased resistance. Similarly when the resistance falls off the speed increases, the balls rise, and G rises shutting off the

steam.

The question we now wish to solve is-Given the speed of revolution, what position will the balls take up ? and after that, supposing the speed change, what force will be exerted to move the slider?

The first of these questions cannot be solved by the use of the principle of work, there being no energy exerted or work done, consequently we must treat it by use of the laws of motion, or at least we will use the result obtained in theoretical mechanics for this case.

Circular Motion-Centrifugal Force. The two balls move in a circle, the plane of which is at right angles to the spindle,

their motion is then one case of circular motion, and hence we will first consider circular motion generally.

In order to cause a body to move at uniform. velocity in a horizontal circle, we

a

Fig. 168.

must do one of two things. Either we must place it

inside a cylinder of the given radius, Fig. 168 (a), or else we must fasten it to a centre O in the plane by a string equal in length to the given radius.

[This latter is not quite accurate, as we shall see a little further on, but is put in this way for simplicity, to keep all the bodies in one plane; also we consider a small particle, whereas in the figure we have given it definite dimensions.]

If now we start the body with a velocity V at right angles to the radius, it will, in the absence of friction, and of resistances generally, continue to move in the circle at this speed.

We can now easily see what is the nature of the actions between the body and the cylinder or string respectively.

The only action which can exist between the body and the cylinder is a direct normal pressure between their surfaces, since there is no friction. But there must be an action of some sort, because otherwise the body would go on moving with a velocity V in the same direction as we started it, ¿.e. along the tangent. Hence then there is a normal pressure, say X, between the body and the cylinder surface, or at any instant the body exerts a push X outwards on the cylinder, and the cylinder exerts a push inwards towards O on the body.

Next for the string, this plainly can exert no effect on the body other than a direct pull; by having a solid rod in place of the string (Fig. 168) and taking hold of it at O, it would be possible to apply a force at right angles to the rod, but it is evidently impossible to do any such a thing with a string, a direct pull along its length is the only possible force the string can apply. The action between body and string then is a mutual pull, the string pulls the body with a force X, and the body pulls the string with a force X also (see page 106). It appears then that a body which is constrained by contact with some other body to move uniformly in a circle, is at every instant exerting on the constraining body a force X,

outward along the radius at the instant. The force is of course constant, since all parts of the path are absolutely identical as regards motion, and therefore also force.

This force X is called the Centrifugal Force; we have seen that it must exist, but its value we will take as known, being determined in treatises on theoretical mechanics. If

The first is generally more convenient.

In dealing with a body of definite dimensions, r is the radius of the circle described by the C. G., and the total centrifugal force is given by putting W = weight of whole body.

We can now return to the consideration of the Governor.

When the whole is steadily rotating in equilibrium there will be no pull on the rods EE (Fig. 167); hence we may, in the first instance, omit them, and thus obtain Fig. 169, where also we only consider one ball, the motions being identical.

h

N

I sin 0

O is the centre of the ball. W its weight.

Scose

Yw.

Fig. 169.

Let

ON=r, AN=h, <NAO=0, A= angular velocity. Since the ball is rotating in a circle at a constant speed some body must be exerting on it a force X along ON, or it must be exerting on the constraining body a force X in the direction NO.

The constraining body is now the rod AO, which does not lie in the plane of the circle (see note on page 248),