GYROSCOPE 353 prolongation of the supporting string. A short analytical investigation of this particular case may serve to give some idea of how problems on the gyroscope generally may be attacked. For clearness in the woodcut (fig. 3) we suppose the ring containing the wheel removed. Let K represent the wheel of mass M; h the distance of the centre of gravity G from ; OC the axis of the wheel, supposed of unit length ; OA, OB, two other axes through the fixed point at right angles to OC and to each other ; OX, OY, OZ, three rect angular axes fixed in space, with which OA, 013, OC (axes movable with the body) initially coincide ; i|/, 0, 6, angles on the unit sphere which define the point C. If A, B, C be the moments of inertia, and -a-, p, a the angular velocities about OA, OB, OC respectively, the equation of kinetic energy T gives but -a sin $ - i// sin cos <p ; p = Q cos $ + i/> sin 8 sin d> ; <rit cos 6 + <p. Substituting and making A = B, as is clearly the case, we have T = ^ Ay 2 + Aip sin -0 + C((J/ cos + <j>) 2 . Applying Lagrange s equations of motion, which are for this case dt, dty f/if/ dt _ d$ dfy
- . ^-lT = M<77tsin0,
dt de de we get (1) ~ AiJ<sin 2 + C(^cos0 + <+>) cos =0; ct t dt = 0; (3) Ay - Aip sin 6 cos 6 + C sin 6^(4> cos + <j>) = My/t sin 0. Integrating (2), we get a constant = 7;,, which represents the angular velocity about OC. Now if the motion be steady we must have both B and i|/ constants. Let them be represented by o and p. respectively. Then equation (3) gives - A,u 2 sin o cos o + C/J.H sin a = M<//t sin a . M<7/i + A,u 2 cos a Ct By this formula we can calculate, for any particular instrument, the angular velocity in azimuth from having given the angular velocity of rotation of the fly-wheel and vice versa. An important case of motion, and one also very interest ing both mathematically and physically, is got by including a gyroscope in a pendulum bob and supporting the rod from a universal flexure joint. This constitutes the gyroscopic pendulum. The joint is usually got by attaching a short length of fine steel wire rigidly to the end of the rod, and suspending the whole by means of the other end of this wire firmly clamped into a fixed support. When the gyroscope is rapidly rotated, and the pendulum drawn a little aside from the vertical and then let go, its lower end is observed to describe a beautiful curve consisting of a series of equal closed loops, all equally near each other and radiating from a centre point. This curve, which is a species of hypotrochoid, is figured in Thomson and Tail s Natural Philosophy, where also the whole theory of the gyroscopic pendulum will be found. As has been already mentioned, a remarkable apparent effect is produced by the earth s diurnal rotation upon a rapidly rotating gyroscope. This arises from what Foucault called the "fixity of tha plane of rotation," and what Thomson and Tait have recently called " gyrostatic domi nation." In virtue of this principle, the axis of the rotating fly-wheel tends always to preserve a fixed direction in space, and, in consequenc3, will appear to move in a direction opposite to that in which the earth s rotation is at each instant actually carrying it. If we suppose the gyroscope represented at fig. 1 to have all de grees of free rotation round the point 0, and to be in every way exactly balanced about that point, and also to have all its pivots nearly void of friction, then, at whatever part of the earth s surface it may be placed, the fly-wheel while rotating rapidly will Le observed to move gradually and finally take up such a position that its axis OC is parallel to the earth s axis, and also that its direction of rotation round OC is the same as that of the earth round its axis. Should the ring K be fixed so that the axis GC can only move in a horizontal plane, then the ring L will move in azimuth till it has placed GC in the direction north and south and such that the direction of rotation of the fly-wheel coincides with that of the earth. Further, should the ring L be fixed in the plane of the meridian, so that GC can only move iu altitude, then GC will be observed to tilt up till it is parallel to the earth s axis, the direction of rotation of the wheel being, as before, the same as that of the earth. These effects may be explained as follows : Let A bo the latitude of the place, and a? the angular velocity of the earth on its axis ; also, at starting, let the ring K be in the horizontal plane, and let OC make an angle a, in azimuth, with a horizontal line ON drawn from northwards ; further, let tho rotation of the wheel round OC be positive. Now ta can be resolved into three angular velocities at right angles to each other, viz. : o> sin A. round the vertical line, ta cos A. cos o round a line parallel to OC, and ca cos A sin a round a line parallel to ON. Of these &>sinA. gives the velocity in azimuth, cos A. cos a can only affect the velocity of the wheel, while co cos A sin a gives the velocity of OC in altitude. These being the actual component velocities communicated to the gyroscope by the earth, the apparent velocities will be equal and opposite to them. The apparatus actually employed by Foucault to demon strate the rotation of the earth differed somewhat from that represented in fig. 1. In it the corresponding ring to L was suspended from a fixed stand by a thread without torsion, and rested at its lowest point by a pivot in an agate cup. Also the ring corresponding to K, which carried the fly-wheel, rested on knife edges within L, and could be removed at pleasure, in order that the rotation might be given to the fly-wheel, Great care was also taken to have every part thoroughly well balanced. It is stated that the experiment was several times successfully performed, and Foucault by means of his apparatus was thus able, without astronomical observation, to find out the latitude of the place, the east and west points, and the rate of the earth s motion. The same experiment has lately, it is said, been successfully performed by Mr G. M. Hopkins. His fly wheel, being driven by electricity, has the advantage of rotating at a uniform rate for any length of time. By attaching a small mirror to the frame which carries the revolving wheel, and using a spot of light reflected from it as an index, he has been able to make manifest the earth s rotation in a very short time. An ingenious practical application of the gyroscope principle was suggested and carried into effect about the year 1856 by Professor Piazzi Smyth. His aim was to devise a telescope-stand which would always remain level on board ship, notwithstanding the pitching and rolling, and so facilitate the taking of astronomical observations at sea. For this purpose the stand was supported on gimbals, and underneath it were placed on fine pivots several heavy fly-wheels which could be put in rapid rotation, some on vertical and some on horizontal axes. The complete appar atus, involving many ingenious details as to driving the fly-wheels, &c., was tested by Professor Smyth on board the yacht "Titania" during a voyage to Teneriffe, and found to work with perfect satisfaction. A full account of the method will be found in Trans. Royal Scottish Society of Arts, vol. iv. GYROSTAT. This is a modification of the gyroscope, devised bySir William Thomson, which has been used by him as well as by Professor Tait for a number of years to illustrate th.3 dynamics of rotating rigid bodies. It consists essentially of a fly-wheel, with a massive rim, fixed on the middle of an axis which can rotate on fine steel pivots inside a rigid case. The rigid case exactly resembles a similarly-shaped, but hollow, fly-wheel and axis closely surrounding the other but still leaving it freedom to move. Slits are made in the
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