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 θ and Aμ sin θ, OA being perpendicular to OC′ in the plane COC′; so that the vector OK has the components
and the horizontal component
CK = OC′ sin θ − C′K cos θ
= G′ sin θ − Aμ sin θ cos θ.
The velocity of K being equal to the impressed couple Oh,
and dropping the factor sin θ,
the condition for steady motion.
Solving this as a quadratic in μ, the roots μ1, μ2 are given by
| μ1, μ2 = | G′ | sec θ [1 ± √ (1 − | 4A2n2 | cos θ) ]; |
| 2A | G′2 |
and the minimum value of G′ = CR for real values of μ is given by
| G′2 | = cos θ, | CR | 2√(cos θ); |
| 4A2n2 | An |
for a smaller value of R the top cannot spin steadily at the inclination θ to the upward vertical.
Interpreted geometrically in fig. 4
so that K lies on a hyperbola with OC, OC′ as asymptotes.
4. Suppose the top or gyroscope, instead of moving freely about the point O, is held in a ring or frame which is compelled Constrained motion of the gyroscope. 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
gMh sin θ − N = μ·CK = sin θ G′μ − Aμ2 cos θ,
= A sin θ cos θ (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
 |
| Fig. 10. |
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.
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.
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 weightCentrifugal machine. 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)
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 θ, about 2312°, in a period of 26,000 years, so that R/μ = 26000 × 365; and the mean couple producing precession is
one 12 millionth part of 12CR2, 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′,
the formula which suffices to explain most gyroscopic action.
Thus a carriage running round a curve experiences, in consequence Gyroscopic action of railway wheels. 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,
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,
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 α, given by
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 α to the vertical, the equation (2) § 4 becomes
Suppose, for instance, that OC is parallel to the earth’s axis, and that the frame is fixed in the meridian; then α 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, α − θ = E,
| tan E = | G′μ sin α | ≈ | G′μ | sin α. |
| gMh + G′μ cos α | gMh |
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′,The barogyroscope. 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.
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.Foucault’s gyroscope.
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.)
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 theoriaeGyroscopic horizon. 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.
The instrument is in favour with French navigators, perfected by
