to the speed of the machine whose work is to be measured; and by means of a dynamometer-spring the length of each stroke of the click is adjusted so as to be proportional to the effort exerted at the time. The result is that the total extent of motion of the ratchet-wheel in a given time is proportional to the work performed. It is obvious that the frictional catch might be applied to this apparatus (Article 197, page 211).

348. Measurement of Friction.—Under the head of Dynamometers may be classed apparatus for the experimental measurement of friction.

If by means of any kind of dynamometer whose use does not involve the interruption of the performance of the ordinary work of a train of mechanism, we measure the power transmitted at two parts of that train, the difference will be the power expended in overcoming the friction of the intermediate parts. Hirn's Pandynamometer (Article 344, page 387) seems well adapted for experiments of this class. The power of a steam engine, as exerted in the cylinder, may be measured by means of the indicator, and the power transmitted to machinery which that engine drives, by a suitable dynamometer; and the difference will be power expended chiefly in overcoming the friction of the inter me liate mechanism.

Special apparatus for measuring the friction of axles is used, not only for purposes of scientific investigation as to the co-efficients of friction of different pairs of surfaces in different states, but for practically testing the lubricating properties of oil and grease. Two forms of apparatus may be described.

I. Statical Apparatus.-A short cylindrical axle, of a convenient diameter (say 2, 3, or 4 inches), is supported at its ends by bearings on the top of a pair of strong fixed standards. The ends of the axle overhang their bearings, and carry a pair of equal and similar pulleys, by means of which it is driven at a speed equal, or nearly equal, to the greatest intended working speed of the axles with which the unguents to be tested are to be used in practice. The object of driving the axle at both ends is to ensure great steadiness of motion. The driving-gear ought to be capable of reversing the direction of rotation. At the middle of its length the axle is turned so as to form a very accurate and smooth journal, of a length equal to from 1 to 2 times its diameter. Upon that journal there hangs a plumber-block or axle-box, fitted with a suitable bush or bearing. That plumber-block is rigidly connected with a heavy mass of suitable material, such as cast iron, so as to form as it were a pendulum hanging from the journal in the middle of the axle, and of a weight suited to produce a pressure on the journal equal to the greatest pressure to which the unguent is to be exposed in practice (see Article 310, page 353). The

pendulum is furnished with an index and graduated arc, to show its deviation from a vertical position.

The hanging plumber-block having been supplied with the unguent to be tested, the axle is to be driven at full speed, first in one direction, and then in the contrary direction, and the two contrary deviations of the pendulum observed. Let e denote the half-sum of those deviations, expressed in circular measure to radius unity; c, the distance from the axis of rotation to the centre of gravity of the pendulum; r, the radius of the journal; let W be the weight of the pendulum; then the mean statical moment of the pendulum is

W ce nearly;

and that moment balances the moment of friction (Article 311, page 356), whose value is f Wr nearly, and will be afterwards shown to be exactly

Wr sin 2,

being the angle of repose. Equating, therefore, those two equal moments, we find

The distance, c, of the centre of gravity of the pendulum from the axis may be found experimentally, by applying a known weight at a known horizontal distance from the axis, so as to make the pendulum deviate, and observing the deviation. Let P be the weight so applied, x its leverage, the deviation which it produces;

then, if there were no friction, we should have

In order to eliminate the effects of friction from the determination of c, the load P with the leverage x should be applied at contrary sides, so as to increase the deviation of the pendulum, while the axle is rotating in the two contrary directions.

Let sin be the mean of the sines of the deviations produced by friction alone, and sin ✪ the mean of the sines of the deviations produced by the friction and the load P together; then we shall have

II. Dynamic or Kinetic Apparatus. To measure the friction of an axle by means of its retarding effect upon a rotating mass, the axle

is supported on suitable bearings at its ends, as in the Statical Apparatus just described; and at the middle of its length it has fitted on it, and accurately balanced, a round disc acting as a flywheel, of weight sufficient to produce the required pressure on the bearings. (See Article 310, page 353.) The numbers of turns made by the axle are counted and indicated by means of a light and easily-driven train of small wheels, with dial-plates and indexes.

The axle is provided with driving-gear of a kind which can be instantly disengaged when required; for example, a fast pulley on one overhanging end, with a loose pulley alongside of it, the loose pulley being carried, not by the fly-axle itself, but by a separate axle in the same straight line with the fly-axle.

After the axle with its fly-disc has been set in motion at a speed greater than the working speed of the axles to which the unguent to be tested is to be applied in practice, the driving-gear is to be disengaged; when the speed of rotation will undergo a gradual retardation through the friction of the journals. The numbers of turns made in a series of equal intervals of time (for example, intervals of thirty seconds, or of sixty seconds, or of a hundred seconds) are to be observed on the counting dials, and noted down. Let W denote the weight of the whole rotating mass, consisting of the axle with its fly-disc; e, the radius of gyration of that mass. (See Article 313, page 357). Let t be the uniform length in seconds of the intervals of time during which the numbers of revolutions are recorded; and in one of those intervals let the disc make n revolutions, and in the next interval n' revolutions. Then the mean angular velocity is,

during the first interval,

2x n'

t

; and treating the rate of retardation as sensibly uniform, the retardation which takes place during the t seconds which elapse from the middle of the first interval to the middle of the second interval is

and to produce that retardation in the course of t seconds in a body' whose moment of inertia is W2, there is required a retarding moment of the following value:

Part of the retarding moment is due to the resistance of the air; but if the fly is a smooth round disc without arms, this may be

neglected for the purpose of the experiments, and the whole moment treated as due to axle-friction. Let r be the radius of the journals, and ƒ the co-efficient of friction: then, as before, the moment of friction is very nearly ƒ W r; and by equating this to the retarding moment, and dividing both sides of the equation by W r, we obtain the following formula for the co-efficient of friction:

When the numbers of revolutions have been observed during a series of more than two equal intervals of time, the formula 2 for the co-efficient of friction is to be applied to each consecutive pair of intervals, and a mean of the results taken.

୧

The radius of gyration e and the radius of the journals r should of course be expressed in the same units of measure. In British measures, feet are the most convenient for the present purpose. The constant factor has the following values :

Similar experiments may be made with a disc rotating about a vertical axis, and supported by a pivot; regard being had to the value of the moment of friction of a pivot, as stated in Article 311, page 356.

To find the square 2 of the radius of gyration by experiment, fix a pair of slender pins in the two faces of the disc at two points opposite each other, and near its circumference; hang up the disc with its axle by these pins, and make it swing like a pendulum in a plane perpendicular to its axis; count the number of single swings in some convenient interval of time; calculate their number per second, and let N denote that number. Then calculate the length L of the equivalent simple pendulum, by the following formula:

The constant factor of this expression, being the length of the seconds pendulum, has approximately the following values:

g
= 3.26 feet
-2

=

0.992 mètre.

.(5.)

Let C be the distance from the point of suspension to the axis of figure of the disc and axle; then the square of the radius of gyration is calculated as follows::

When the object of the experiments is not to obtain absolute values of the co-efficient of friction, but merely to compare one specimen of unguent with another, it is sufficient to compare together the rates of retardation with the two unguents in equal intervals of time.

III. Comparison of Heating Effects.-For the same purpose of comparing unguents with each other, without measuring the friction absolutely, the heating effects of the friction with different unguents are sometimes compared together. The apparatus used is similar to that described under the head of (I.) Static Apparatus; except that there is no reversing-gear, and that the pendulum, or loaded plumberblock, has no index nor graduated arc, and is provided with a thermometer, having its bulb immersed in the passage through which the unguent flows from the grease-box to the journal. Another thermometer, hung on the wall of the room, shows the temperature of the air. The axle is driven at its proper speed, until the temperature shown by the first-mentioned thermometer ceases to rise; and then the elevation of that temperature above the temperature of the air is noted. (See Article 310, page 353.)

In all experiments for the purpose of comparing unguents with each other, care should be taken to remove one sort of unguent completely from the rubbing surfaces, grease-box, and passages, before beginning to test the effect of another sort, lest the mixture of different sorts of unguents should make the experiments inconclusive.

ADDENDUM TO ARTICLE 309, PAGE 348.

Friction of Pistons and Plungers. From experiments made by Mr. William More and others, it appears that the friction of ordinary pistons and plungers may be estimated at about onetenth of the amount of the effective pressure exerted by the fluid on the piston.

The friction of a plunger working through a cupped leather collar is equal to the pressure of the fluid upon a ring equal in circumference to the collar, and of a breadth which, according to Mr. More's experiments, is about 0·4 of the depth of bearing-surface of the collar; and according to the experiments of Messrs. Hick and Luthy, from 01 to 015 inch (= from 25 to 375 millimètres), according to the state of lubrication of the collar.