Part 9

Pure gypsum is colourless or white, but it is often tinted, especially in the alabaster variety, grey, yellow or pink. Gypsum crystallizes with two molecules of water, equal to about 21% by weight, and consequently has the formula CaSO4.2H2O. By exposure to strong heat all the water may be expelled, and the substance then has the composition of anhydrite (q.v.). When the calcination, however, is conducted at such a temperature that only about 75% of the water is lost, it yields a white pulverulent substance, known as "plaster of Paris," which may readily be caused to recombine with water, forming a hard cement. The gypsum quarries of Montmartre, in the north of Paris, were worked in Tertiary strata, rich in fossils. Gypsum is largely quarried in England for conversion into plaster of Paris, whence it is sometimes known as "plaster stone," and since much is sent to the Staffordshire potteries for making moulds it is also termed "potter's stone." The chief workings are in the Keuper marls near Newark in Nottinghamshire, Fauld in Staffordshire and Chellaston in Derbyshire. It is also worked in Permian beds in Cumberland and Westmorland, and in Purbeck strata near Battle in Sussex.

Gypsum frequently occurs in association with rock-salt, having been deposited in shallow basins of salt water. Much of the calcium in sea-water exists as sulphate; and on evaporation of a drop of sea-water under the microscope this sulphate is deposited as acicular crystals of gypsum. In salt-lagoons the deposition of the gypsum is probably effected in most cases by means of micro-organisms. Waters containing sulphuretted hydrogen, on exposure to the air in the presence of limestone, may yield gypsum by the formation of sulphuric acid and its interaction with the calcium carbonate. In volcanic districts gypsum is produced by the action of sulphuric acid, resulting from the oxidation of sulphurous vapours, on lime-bearing minerals, like labradorite and augite, in the volcanic rocks: hence gypsum is common around solfataras. Again, by the oxidation of iron-pyrites and the action of the resulting sulphuric acid on limestone or on shells, gypsum may be formed; whence its origin in most clays. Gypsum is also formed in some cases by the hydration of anhydrite, the change being accompanied by an increase of volume to the extent of about 60%. Conversely gypsum may, under certain conditions, be dehydrated or reduced to anhydrite.

Some of the largest known crystals of selenite have been found in southern Utah, where they occur in huge geodes, or crystal-lined cavities, in deposits from the old salt-lakes. Fine crystals, sometimes curiously bent, occur in the Permian rocks of Friedrichroda, near Gotha, where there is a grotto called the Marienglashöhle, close to Rheinhardsbrunn. Many of the best localities for selenite are in the New Red Sandstone formation (Trias and Permian), notably the salt-mines of Hall and Hallein, near Salzburg, and of Bex in Switzerland. Excellent crystals, usually of a brownish colour arranged in groups, are often found in the brine-chambers and the launders used in salt-works. Selenite also occurs in fine crystals in the sulphur-bearing marls of Girgenti and other Sicilian localities; whilst in Britain very bold crystals are yielded by the Kimeridge clay of Shotover Hill near Oxford. Twisted crystals and rosettes of gypsum found in the Mammoth Cave, Kentucky, have been called "oulopholites" ([Greek: oulos], "woolly"; [Greek: phôleos], "cave").

In addition to the use of gypsum in cement-making, the mineral finds application as an agricultural agent in dressing land, and it has also been used in the manufacture of porcelain and glass. Formerly it was employed, in the form of thin cleavage-plates, for glazing windows, and seems to have been, with mica, called _lapis specularis_. It is still known in Germany as _Marienglas_ and _Fraueneis_. Delicate cleavage-plates of gypsum are used in microscopic petrography for the determination of certain optical constants in the rock-forming minerals. (F. W. R.*)

GYROSCOPE AND GYROSTAT. These are scientific models or instruments designed to illustrate experimentally the dynamics of a rotating body such as the spinning-top, hoop and bicycle, and also the precession of the equinox and the rotation of the earth.

The gyroscope (Gr. [Greek: gyros], ring, [Greek: skopein], to see) may be distinguished from the gyrostat ([Greek: gyros], and [Greek: statikos], stationary) as an instrument in which the rotating wheel or disk is mounted in gimbals so that the principal axis of rotation always passes through a fixed point (fig. 1). It can be made to imitate the motion of a spinning-top of which the point is placed in a smooth agate cup as in Maxwell's dynamical top (figs. 2, 3). (_Collected Works_, i. 248.) A bicycle wheel, with a prolongation of the axle placed in a cup, can also be made to serve (fig. 4).

[Illustration: FIG. 1.]

[Illustration: FIG. 2.]

The gyrostat is an instrument designed by Lord Kelvin (_Natural Philosophy_, § 345) to illustrate the more complicated state of motion of a spinning body when free to wander about on a horizontal plane, like a top spun on the pavement, or a hoop or bicycle on the road. It consists essentially of a massive fly-wheel concealed in a metal casing, and its behaviour on a table, or with various modes of suspension or support, described in Thomson and Tait, _Natural Philosophy_, serves to illustrate the curious reversal of the ordinary laws of statical equilibrium due to the _gyrostatic domination_ of the interior invisible fly-wheel, when rotated rapidly (fig. 5).

The toy shown in figs. 6 and 7, which can be bought for a shilling, is acting as a gyroscope in fig. 6 and a gyrostat in fig. 7.

[Illustration: FIG. 3.]

[Illustration: FIG. 4.]

The gyroscope, as represented in figs. 2 and 3 by Maxwell's dynamical top, is provided with screws by which the centre of gravity can be brought into coincidence with the point of support. It can then be used to illustrate Poinsot's theory of the motion of a body under no force, the gyroscope being made kinetically unsymmetrical by a setting of the screws. The discussion of this movement is required for Jacobi's theorems on the allied motion of a top and of a body under no force (Poinsot, _Théorie nouvelle de la rotation des corps_, Paris, 1857; Jacobi, _Werke_, ii. Note B, p. 476).

To imitate the movement of the top the centre of gravity is displaced from the point of support so as to give a preponderance. When the motion takes place in the neighbourhood of the downward vertical, the bicycle wheel can be made to serve again mounted as in fig. 8 by a stalk in the prolongation of the axle, suspended from a universal joint at O; it can then be spun by hand and projected in any manner.

[Illustration: FIG. 5.]

[Illustration: FIG. 6.]

[Illustration: FIG. 7.]

[Illustration: FIG. 8.]

[Illustration: FIG. 9.]

The first practical application of the gyroscopic principle was invented and carried out (1744) by Serson, with a spinning top with a polished upper plane surface for giving an artificial horizon at sea, undisturbed by the motion of the ship, when the real horizon was obscured. The instrument has been perfected by Admiral Georges Ernest Fleuriais (fig. 9), and is interesting theoretically as showing the correction required practically for the rotation of the earth. Gilbert's barogyroscope is devised for the same purpose of showing the earth's rotation; a description of it, and of the latest form employed by Föppl, is given in the _Ency. d. math. Wiss._, 1904, with bibliographical references in the article "Mechanics of Physical Apparatus." The rotation of the fly-wheel is maintained here by an electric motor, as devised by G. M. Hopkins, and described in the _Scientific American_, 1878. To demonstrate the rotation of the earth by the constancy in direction of the axis of a gyroscope is a suggestion that has often been made; by E. Sang in 1836, and others. The experiment was first carried out with success by Foucault in 1851, by a simple pendulum swung in the dome of the Pantheon, Paris, and it has been repeated frequently (_Mémoires sur le pendule_, 1889).

A gyroscopic fly-wheel will preserve its original direction in space only when left absolutely free in all directions, as required in the experiments above. If employed in steering, as of a torpedo, the gyroscope must act through the intermediary of a light relay; but if direct-acting, the reaction will cause precession of the axis, and the original direction is lost.

The gyrostatic principle, in which one degree of freedom is suppressed in the axis, is useful for imparting steadiness and stability in a moving body; it is employed by Schlick to mitigate the rolling of a ship and to maintain the upright position of Brennan's monorail car.

Lastly, as an application of gyroscopic theory, a stretched chain of fly-wheels in rotation was employed by Kelvin as a mechanical model of the rotary polarization of light in an electromagnetic field; the apparatus may be constructed of bicycle wheels connected by short links, and suspended vertically.

_Theory of the Symmetrical Top._

1. The physical constants of a given symmetrical top, expressed in C.G.S. units, which are employed in the subsequent formulae, are denoted by M, h, C and A. M is the weight in grammes (g) as given by the number of gramme weights which equilibrate the top when weighed in a balance; h is the distance OG in centimetres (cm.) between G the centre of gravity and O the point of support, and Mh may be called the preponderance in g.-cm.; Mh and M can be measured by a spring balance holding up in a horizontal position the axis OC in fig. 8 suspended at O. Then gMh (dyne-cm. or ergs) is the moment of gravity about O when the axis OG is horizontal, gMh sin [theta] being the moment when the axis OG makes an angle [theta] with the vertical, and g = 981 (cm./s²) on the average; C is the moment of inertia of the top about OG, and A about any axis through O at right angles to OG, both measured in g-cm.².

To measure A experimentally, swing the top freely about O in small plane oscillation, and determine the length, l cm., of the equivalent simple pendulum; then

(1) l = A/Mh, A = Mhl.

Next make the top, or this simple pendulum, perform small conical revolutions, nearly coincident with the downward vertical position of equilibrium, and measure n, the mean angular velocity of the conical pendulum in radians / second; and T its period in seconds; then

(2) 4[pi]²/T² = n² = g/l = gMh/A;

and f = n/2[pi] is the number of revolutions per second, called the _frequency_, T = 2[pi]/n is the period of a revolution, in seconds.

Steady motion of the top.

2. In the popular explanation of the steady movement of the top at a constant inclination to the vertical, depending on the composition of angular velocity, such as given in Perry's _Spinning Tops_, or Worthington's _Dynamics of Rotation_, it is asserted that the moment of gravity is always generating an angular velocity about an axis OB perpendicular to the vertical plane COC´ through the axis of the top OC´; and this angular velocity, compounded with the resultant angular velocity about an axis OI, nearly coincident with OC´, causes the axes OI and OC´ to keep taking up a new position by moving at right angles to the plane COC´, at a constant precessional angular velocity, say µ rad./sec., round the vertical OC (fig. 4).

If, however, the axis OC´ is prevented from taking up this precessional velocity, the top at once falls down; thence all the ingenious attempts--for instance, in the swinging cabin of the Bessemer ship--to utilise the gyroscope as a mechanical directive agency have always resulted in failure (_Engineer_, October 1874), unless restricted to actuate a light relay, which guides the mechanism, as in steering a torpedo.

An experimental verification can be carried out with the gyroscope in fig. 1; so long as the vertical spindle is free to rotate in its socket, the rapidly rotating wheel will resist the impulse of tapping on the gimbal by moving to one side; but when the pinch screw prevents the rotation of the vertical spindle in the massive pedestal, this resistance to the tapping at once disappears, provided the friction of the table prevents the movement of the pedestal; and if the wheel has any preponderance, it falls down.

Familiar instances of the same principles are observable in the movement of a hoop, or in the steering of a bicycle; it is essential that the handle of the bicycle should be free to rotate to secure the stability of the movement.

The bicycle wheel, employed as a spinning top, in fig. 4, can also be held by the stalk, and will thus, when rotated rapidly, convey a distinct muscular impression of resistance to change of direction, if brandished.

Elementary demonstration of the condition of steady motion.

3. A demonstration, depending on the elementary principles of dynamics, of the exact conditions required for the axis OC´ of a spinning top to spin steadily at a constant inclination [theta] to the vertical OC, is given here before proceeding to the more complicated question of the general motion, when [theta], the inclination of the axis, is varying by nutation.

It is a fundamental principle in dynamics that if OH is a vector representing to scale the angular momentum of a system, and if Oh is the vector representing the axis of the impressed couple or torque, then OH will vary so that the velocity of H is represented to scale by the impressed couple Oh, and if the top is moving freely about O, Oh is at right angles to the vertical plane COC´, and

(1) Oh = gMh sin [theta].

In the case of the steady motion of the top, the vector OH lies in the vertical plane COC´, in OK suppose (fig. 4), and has a component OC = G about the vertical and a component OC´ = G´, suppose, about the axis OC; and G´ = CR, if R denotes the angular velocity of the top with which it is spun about OC´.

If µ denotes the constant precessional angular velocity of the vertical plane COC´ the components of angular velocity and momentum about OA are µ sin [theta] and Aµ sin [theta], OA being perpendicular to OC´ in the plane COC´; so that the vector OK has the components

(2) OC´ = G´, and C´K = Aµ sin[theta],

and the horizontal component

(3) CK = OC´ sin[theta] - C´K cos[theta] = G´ sin[theta] - Aµ sin[theta] cos[theta].

The velocity of K being equal to the impressed couple Oh,

(4) gMh sin[theta] = µ·CK = sin [theta] (G´µ - Aµ² cos [theta]),

and dropping the factor sin[theta],

(5) Aµ² cos[theta] - G´µ + gMh = 0, or Aµ² cos[theta] - CRµ + An² = 0,

the condition for steady motion.

Solving this as a quadratic in µ, the roots µ1, µ2 are given by _ _ G´ | 4A²n² | (6) µ1, µ2 = -- sec [theta] |1 ± [root] (1 - ----- cos [theta]) |; 2A |_ G´² _|

and the minimum value of G´ = CR for real values of µ is given by

G´² CR (7) ----- = cos [theta], -- = 2[root](cos [theta]); 4A²n² An

for a smaller value of R the top cannot spin steadily at the inclination [theta] to the upward vertical.

Interpreted geometrically in fig. 4

(8) µ = gMh sin [theta]/CK = An²/KN, and µ = C´K/A sin [theta] = KM/A,

(9) KM·KN = A²n²,

so that K lies on a hyperbola with OC, OC´ as asymptotes.

Constrained motion of the gyroscope.

4. Suppose the top or gyroscope, instead of moving freely about the point O, is held in a ring or frame which is compelled to rotate about the vertical axis OC with constant angular velocity µ; then if N denotes the couple of reaction of the frame keeping the top from falling, acting in the plane COC', equation (4) § 3 becomes modified into

(1) gMh sin [theta] - N = µ·CK = sin [theta] G´µ - Aµ² cos [theta],

(2) N = sin [theta] (Aµ² cos [theta] - G´µ + gMh) = A sin[theta] cos [theta] (µ - µ1)(µ - µ2);

and hence, as µ increases through µ2 and µ1, the sign of N can be determined, positive or negative, according as the tendency of the axis is to fall or rise.

When G´ = CR is large, µ2 is large, and

(3) µ1 [~=] gMh/G´ = An²/CR,

the same for all inclinations, and this is the precession observed in the spinning top and centrifugal machine of fig. 10 This is true accurately when the axis OC' is horizontal, and then it agrees with the result of the popular explanation of § 2.

[Illustration: FIG. 10.]

If the axis of the top OC´ is pointing upward, the precession is in the same direction as the rotation, and an increase of µ from µ1 makes N negative, and the top rises; conversely a decrease of the procession µ causes the axis to fall (Perry, _Spinning Tops_, p. 48).

If the axis points downward, as in the centrifugal machine with upper support, the precession is in the opposite direction to the rotation, and to make the axis approach the vertical position the precession must be reduced.

Centrifugal machine.

This is effected automatically in the Weston centrifugal machine (fig. 10) used for the separation of water and molasses, by the friction of the indiarubber cushions above the support; or else the spindle is produced downwards below the drum a short distance, and turns in a hole in a weight resting on the bottom of the case, which weight is dragged round until the spindle is upright; this second arrangement is more effective when a liquid is treated in the drum, and wave action is set up (_The Centrifugal Machine_, C. A. Matthey).

Similar considerations apply to the stability of the whirling bowl in a cream-separating machine.

We can write equation (1)

(4) N = An² sin [theta] - µ·CK = (A^n² - KM·KN) sin [theta]/A,

so that N is negative or positive, and the axis tends to rise or fall according as K moves to the inside or outside of the hyperbola of free motion. Thus a tap on the axis tending to hurry the precession is equivalent to an impulse couple giving an increase to C´K, and will make K move to the interior of the hyperbola and cause the axis to rise; the steering of a bicycle may be explained in this way; but K1 will move to the exterior of the hyperbola, and so the axis will fall in this second more violent motion.

Friction on the point of the top may be supposed to act like a tap in the direction opposite to the precession; and so the axis of a top spun violently rises at first and up to the vertical position, but falls away again as the motion dies out. Friction considered as acting in retarding the rotation may be compared to an impulse couple tending to reduce OC´, and so make K and K1 both move to the exterior of the hyperbola, and the axis falls in both cases. The axis may rise or fall according to the direction of the frictional couple, depending on the shape of the point; an analytical treatment of the varying motion is very intractable; a memoir by E. G. Gallop may be consulted in the _Trans. Camb. Phil. Soc._, 1903.

The earth behaves in precession like a large spinning top, of which the axis describes a circle round the pole of the ecliptic of mean angular radius [theta], about 23½°, in a period of 26,000 years, so that R/µ = 26000 × 365; and the mean couple producing precession is

(5) CRµ sin [theta] = CR² sin 23½°/(26000 × 365),

one 12 millionth part of ½CR², the rotation energy of the earth.

5. If the preponderance is absent, by making the C·G coincide with O, and if Aµ is insensible compared with G´,

(1) N = -G´µ sin [theta],

the formula which suffices to explain most gyroscopic action.

Gyroscopic action of railway wheels.

Thus a carriage running round a curve experiences, in consequence of the rotation of the wheels, an increase of pressure Z on the outer track, and a diminution Z on the inner, giving a couple, if a is the gauge,

(2) Za = G´µ,

tending to help the centrifugal force to upset the train; and if c is the radius of the curve, b of the wheels, C their moment of inertia, and v the velocity of the train,

(3) µ = v/c, G´ = Cv/b,

(4) Z = Cv²/abc (dynes),

so that Z is the fraction C/Mab of the centrifugal force Mv²/c, or the fraction C/Mh of its transference of weight, with h the height of the centre of gravity of the carriage above the road. A Brennan carriage on a monorail would lean over to the inside of the curve at an angle [alpha], given by

(6) tan [alpha] = G´µ/gMh = G´v/gMhc.

The gyroscopic action of a dynamo, turbine, and other rotating machinery on a steamer, paddle or screw, due to its rolling and pitching, can be evaluated in a similar elementary manner (Worthington, _Dynamics of Rotation_), and Schlick's gyroscopic apparatus is intended to mitigate the oscillation.

6. If the axis OC in fig. 4 is inclined at an angle [alpha] to the vertical, the equation (2) § 4 becomes

(1) N = sin [theta] (Aµ² cos [theta] - G´µ) + gMh sin ([alpha] - [theta]).

Suppose, for instance, that OC is parallel to the earth's axis, and that the frame is fixed in the meridian; then [alpha] is the co-latitude, and µ is the angular velocity of the earth, the square of which may be neglected; so that, putting N = 0, [alpha] - [theta] = E,

(2) gMh sin E - G´µ sin ([alpha] - E) = 0,

G´µ sin [alpha] G´µ (3) tan E = --------------------- [~=] --- sin [alpha]. gMh + G´µ cos [alpha] gMh

The barogyroscope.

This is the theory of Gilbert's barogyroscope, described in Appell's _Mécanique rationnelle_, ii. 387: it consists essentially of a rapidly rotated fly-wheel, mounted on knife-edges by an axis perpendicular to its axis of rotation and pointing east and west; spun with considerable angular momentum G´, and provided with a slight preponderance Mh, it should tilt to an angle E with the vertical, and thus demonstrate experimentally the rotation of the earth.

Foucault's gyroscope.

In Foucault's gyroscope (_Comptes rendus_, 1852; Perry, p. 105) the preponderance is made zero, and the axis points to the pole, when free to move in the meridian.

Generally, if constrained to move in any other plane, the axis seeks the position nearest to the polar axis, like a dipping needle with respect to the magnetic pole. (_A gyrostatic working model of the magnetic compass_, by Sir W. Thomson. British Association Report, Montreal, 1884. A. S. Chessin, St Louis Academy of Science, January 1902.)

Gyroscopic horizon.

A spinning top with a polished upper plane surface will provide an artificial horizon at sea, when the real horizon is obscured. The first instrument of this kind was constructed by Serson, and is described in the _Gentleman's Magazine_, vol. xxiv., 1754; also by Segner in his _Specimen theoriae turbinum_ (Halae, 1755). The inventor was sent to sea by the Admiralty to test his instrument, but he was lost in the wreck of the "Victory," 1744. A copy of the Serson top, from the royal collection, is now in the Museum of King's College, London. Troughton's Nautical Top (1819) is intended for the same purpose.