It is the angular velocity which is visible to the eye, and its vector as giving a line of instantaneous rest, shown by a coloured card on the axle of the Maxwell top; but for complete dynamical treatment the A.M. is of greater importance.

The most general motion of the axle would lead too far, where it makes nutations, and describes a path, either undulating or looped or cusped; realised easily with the apparatus of Fig. 42. A condensed treatment will be found in Notes on Dynamics, p. 200, and here are two cases which lead to an algebraical solution of simple character.

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I. Hold the axle up horizontal, and, with no spin of the wheel, project the axle horizontally; the motion is similar to a spherical pendulum.

II. Spin the wheel and hold the axle up above the horizontal, so that when let fall it starts from a cusp and reaches the horizontal, and rises again to a cusp, and so continues in a succession.

But the general case may prove of very complicated character.

The lecture concludes with a digression on the simple principles of Linear Dynamics; the transcription should go on to a single sheet of paper, but it is all the dynamical theory required for a large number of familiar problems, such as those given in Notes on Dynamics (Wyman).

DIGRESSION ON LINEAR DYNAMICS.

A wheeled carriage, electric tram, motor car, or motibus (Fig. 47), W tons, acquires velocity v f/s in t secs through s ft from rest, propelled by a constant force F tons.

ย

g

In a field of gravity, g f/s2, v is acquired in falling freely

The car moves as if disturbed by a horizontal field, diluted to

and Newton's Second Law of Motion is translated by equation (27). If break resistance B tons brings the car to rest in t' secs and s' ft,

line for the middle part of a run at full speed v; completed where the car comes to a stop.

By giving the energy line a slope of F in W, it will represent the level of apparent gravity to a passenger, perpendicular to the plumb line.

A sudden change in the plumb line will represent the jerk, as at stopping and starting.

A passenger walking out at the front feels the floor sloping down and leans back; the jerk of gravity restored tends to throw

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him on his back. Vice versa for leaving at the rear, also at starting, and the change from uniform velocity, as felt by the straphanger; although this is smoothed down in practice, as in actual running the transition corners are smoothed down, and not so noticeable.

Dead resistance may be allowed for by supposing the road slightly up hill at the angle of repose; and no essential alteration is required, except in taking a little off F.

What, for example, is the time gained by cutting out a station on the tube railway? It is the halt and half the time of

the stop and start.

And the time curve is the graph of a Bradshaw Time Table. No mention has been made of the acceleration of the car; the idea is difficult and it is proverbial the engine driver cannot grasp it. But the passenger feels it as a tangible sensation, especially on leaving the car at one end or the other.

Acceleration appears, however, in g; and we postulate the theorems that a falling body will acquire

And the proper place for g is below v and v2; it must not go astray under W.

The gravitation measure of force is used, suitable for dynamical questions in the field of gravity in which we live, and employed universally by the engineer.

A second lecture, of one minute, would carry on with the Statics of the alteration of trim of the floor on the springs, due to passengers entering or leaving, and as the carriage is accelerated or retarded.

The Law of the Spring is assumed as an experimental fact, based on Hooke's vague statement of the law

Ut tensio sic vis.

A third lecture could be devoted to the simple pendulum, and the length required to beat time with the oscillation of a carriage body on the springs, vertical, pitching and rolling.

Thus the vertical oscillation should synchronize with a pendulum of length equal to the set of the springs, the vertical distance the carriage body sinks down on superposition; the law of the spring being supposed to hold.

This is verified with a spring balance and a weight, a 32-lb shot, provided the scale can be graduated uniformly.

But the logical and simple statement of the formula for the beat of the pendulum is

if L denotes the pendulum length which beats the second; as it is L which is determined experimentally, and g is derived from it by the relation

The rolling and pitching oscillation of the carriage body on the springs would introduce the idea of angular inertia, with which this lecture began; this is shown in its simplest form in the carriage wheels, in adding to the linear inertia of the carriage. The measurement of moment of inertia, or second moment, would run into a fourth lecture, provided we had the unlimited time. at the disposal of Marchis in his Sorbonne lectures.

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LECTURE V

THE SCREW PROPELLER

THERE is no exact theory, it must be conceded, of universal acceptance for the screw propeller, and reliance is placed chiefly on an empirical factor based on experience and model experiment, employed in a formula which satisfies the condition of mechanical similitude, so as to predict from a small scale experiment the performance to be expected of the full size machine.

A rational theory can be given of a hydraulic machine or turbine, when the water is compelled to follow a definite path; but where the fluid, air or water, is free to take its own course, as in the screw propeller, no exact treatment is possible until the stream lines have been determined.

Where the screw works in open water or air, the stream line is free to take a line of least action, and the shape is influenced to a great extent by the hull and fixtures in the neighbourhood, and the relative position of the propeller, effects which cannot be considered in a single formula.

Numerous theories will be found in the Abstracts of the Report of the Aeronautical Committee due to various experimenters, and one initial difficulty is to reconcile the conflicting notation employed by each writer; it is time this notation was standardised.

But the formulas are found to be in general agreement in making the thrust T proportional to