# An elementary treatise on rigid dynamics

AN ELEMENTARY TREATISE ON RIGID DYNAMICS

BY W. J. LOUDON
DEMONSTRATOR IN PHYSICS IN THE UNIVERSITY OF TORONTO

NEW YORK; THE MACMILLAN COMPANY; 1896

An elementary treatise on rigid dynamics

PREFACE

This elementary treatise on Rigid Dynamics has arisen out of a course of lectures delivered by me, during the past few years, to advanced classes in the University.

It is intended as a text-book for those who, having already mastered the elements of the Calculus and acquired some familiarity with the methods of Particle Dynamics, wish to become acquainted with the principles underlying the equations of motion of a solid body.

Although indebted to the exhaustive works of Routh and Price for many suggestions and problems, I believe that the arrangement of the work, method of treatment, and more particularly the illustrations, are entirely new and original; and that they will not only aid beginners in appreciating fundamental truths, but will also point out to them the road along which they must travel in order to become intimate with those higher complex motions of a material system which have their culminating point in the region of Physical Astronomy.

My thanks are due Mr. J. C. Glashan of Ottawa, who has kindly read the proofsheets and supplied me with a large collection of miscellaneous problems.

CONTENTS

CHAPTER I
-    Moments of Inertia
-    Illustrative Examples

CHAPTER II
-    Ellipsoids of Inertia
-    Illustrative Examples
-    Equimomental Systems
-    Principal Axes
-    Illustrative Examples

CHAPTER III
-    D'Alembert's Principle
-    Impulsive Equations of Motion
-    Illustrative Examples
-    The Principle of Energy
-    Illustrative Examples

CHAPTER IV
-    Motion about a Fixed Axis. Finite Forces
-    The Pendulum
-    Illustrative Examples
-    Determination of g by the Pendulum
-    Pressure on Fixed Axis
-    Illustrative Examples

CHAPTER V.
-    Motion about a Fixed Axis. Impulsive Forces
-    Centre of Percussion
-    Illustrative Examples
-    Initial Motions. Changes of Constraint
-    Illustrative Examples
-    The Ballistic Pendulum

CHAPTER VI
-    Motion about a Fixed Point. Finite Forces
-    Angular Velocity
-    General Equations of Motion
-    Equations of Motion referred to Axes fixed in Space
-    Euler's Equations of Motion
-    Angular Coordinates of the Body
-    Pressure on the Fixed Point
-    Illustrative Examples
-    Top spinning on a Rough Horizontal Plane
-    Top spinning with Great Velocity on a Rough Horizontal Plane
-    The Gyroscope moving in a Horizontal Plane about a Fixed Point

CHAPTER VII.
-    Motion about a Fixed Point. Impulsive Forces
-    Illustrative Examples

CHAPTER VIII.
-    Motion about a Fixed Point. No Forces acting

CHAPTER IX.
-    Motion of a Free Body
-    Illustrative Examples
-    Impulsive Actions
-    Illustrative Examples

CHAPTER X.
-    The Gyroscope
-    To prove the Rotation of the Earth upon its Axis
-    Hopkins' Electrical Gyroscope
-    Fessel’s Gyroscope
-    Gyroscope of Gustav Magnus
-    Note on the Pendulum and the Top
-    Miscellaneous Examples

CHAPTER X - THE GYROSCOPE

This instrument, to which reference has already been made in connection with motion about a fixed point, consists essentially of a wheel which is put in rotation within an outer ring: the latter being provided with knife edges and other arrangements whereby the whole mass may be experimented upon while the wheel is kept in motion.

A type of gyroscope, known as Foucaults, is shown in Fig. 63, and also more in detail in Figs. 65 and 66.

It is made of a disc, turned to offer the least resistance to the air, which can be made to rotate with great speed (from two hundred and fifty to five hundred times per second) about an axis through its centre of gravity.

This is done by means of the wheelwork motor (driven by hand) shown in Fig. 64, which is geared .up at the top to the small toothed cog-wheel seen in Fig. 63, at the left-hand side of the disc, on the axis of the gyroscope, and within the outer ring.

The axis of rotation is of course movable in the outer ring, and this latter is provided with two knife edges which should be exactly in the prolongation of a line passing through the centre of gravity and perpendicular to the rotation axis.

Four movable masses, two within the ring, and two outside, Fig. 65, are used to adjust the instrument in two perpendicular planes, so that the centre of gravity of the system will be in the line of the knife edges.

It is quite a difficult matter to perform this adjustment, which must be exact; since the slightest deviation of the position of the centre of gravity from this line destroys the value of the results obtained in the pendulum experiment.

The readiest way to adjust the gyroscope is to let it oscillate, under the action of gravity, about the knife edges, the centre of gravity being arranged at first to fall below the line of the knife edges (by properly altering the positions of the movable masses); and then, by slight variations of these positions, to bring the centre, of gravity up until the oscillations about the knife-edge axis are made in from eight to ten seconds: the line of the knife edges is in that case infinitely close to the centre of gravity and the equilibrium nearly neutral.

The Gyroscope moving in a Horizontal Plane about a Fixed Point.

The gyroscope being adjusted, the experiment indicated by the theory of Art. 72 may easily be performed.

It is only necessary to place the instrument on top of the motor so that the wheels are properly geared, and to set the disc in rapid rotation, taking care that the bearings are care fully cleaned and oiled.

Then, placing it as shown in Fig. 65, so that a small pointed hook which is directly in the prolongation of the axis of rotation rests on a little agate cup at the top of an upright stand, the instrument is given a slight angular displacement bodily about a vertical axis passing through the point at which the hook rests, and it slowly moves about the vertical with an angular velocity equal to that found by the theory of Art. 72.

Moreover, the direction of motion is as shown in Fig. 66; that is, the gyroscope moves bodily about a vertical axis (when viewed from above) in the same direction as the disc rotates when viewed by an observer looking towards the fixed point about which the motion takes place.

Thus there is a perfect accord between theory and experiment, and the truth of the fundamental equations of motion is established.

It may be observed also that if the gyroscope be given no initial impulse, but be merely let drop, it will act in the same manner as a top, and oscillate up and down while it keeps in motion about the vertical.

To prove the Rotation of the Earth upon its Axis.

This experiment depends on the permanency of the rotation axis in space.

A stand with pendulum is arranged as shown in Fig. 67.

There is a ring suspended by means of a fibre without torsion from a hook above, and the whole being carefully levelled so that the line of suspension is vertical, the gyroscope is put in rapid rotation and placed in the ring with the knife edges resting within beds provided for them : the ring, being then released by the small screw seen at the right, is quite free in space, and owing to the rapid rotation of the disc the axis of rotation is a permanent axis and remains fixed in space.

Hence, while the earth moves along, carrying with it the stand and observer, the gyroscope preserves its position in space for some time; and if a long index be attached to it in prolongation of the rotation axis or parallel to it, this index will have an apparent motion from east to west, as the observer is carried along with the earth from west to east.

If the pendulum with the gyroscope were placed at the north pole, it is evident that the apparent motion of the index would be 360 in twenty-four hours.

At the equator there would be no apparent motion ; as although a permanent axis would still exist, the earth would simply carry the whole instrument bodily about the rotation axis of the earth.

Electrical Gyroscope.

The defect of Foucault s gyroscope being that it does not keep up its motion long enough to give marked results in the pendulum experiment, an electrical gyroscope has been devised by Mr. Hopkins, who gives a description of his instrument in the Scientific American of July 6, 1878, and also in his recent text book on Physics. His instrument is shown in Fig. 70.

The rectangular frame which contains the wheel is supported by a fine and very hard steel point, which rests upon an agate step in the bottom of a small iron cup at the end of the arm that is supported by the standard. The carefully made steel points, and upon it are placed two cams, one at each end, which operate the current-breaking springs.

The horizontal sides of the frame are of brass, and the vertical sides are iron. To the vertical sides are attached the cores of the electro-magnets. There are two helices and two cores on each side of the wheel, and the wheel has attached to it two armatures, one on each side, which are arranged at right angles to each other. The two magnets are oppositely
arranged in respect of polarity, to render the instrument astatic.

An insulated stud projects from the middle of the lower end of the frame to receive an index that extends nearly to the periphery of the circular base piece and moves over a graduated semicircular scale. An iron point projects from the insulated stud into a mercury cup in the centre of the base piece, and is in electrical communication with the platinum pointed screws of the current breakers. The current-breaking springs are con nected with the terminals of the magnet wires, and the magnets are in electrical communication with the wheel-supporting frame. One of the binding posts is connected by a wire with the mercury in the cup, and the other is connected with the standard. A drop of mercury is placed in the cup that contains the agate step to form an electrical connection between the iron cup and the pointed screw.

The current breaker is contrived to make and break the current at the proper instant, so that the full effect of the mag nets is realized, and when the binding posts are connected with four or six Bunsen cells the wheel rotates at a high velocity.

The wheel will maintain its plane of rotation, and when it is brought into the plane of the meridian, the index will appear to move slowly over the scale in a direction contrary to the earths rotation, but in reality the earth and the scale with it move from west to east, while the index remains nearly stationary.