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From this we see that the value of increases gradually from
715. It may be useful, for special applications, to indicate briefly
where R is the radius of the earth (supposed spherical) in feet, and o its mean density, expressed in terms of the unit just defined. Taking 20,900,000 feet as the value of R, we have o=0.000000368=3.68 x 10-7.
(6) As the mean density of the earth is somewhere about 5.5 times that of water, the density of water in terms of our present unit is
716. The fourth column of the table above gives the time of rotation in seconds, corresponding to each value of the eccentricity, p being assumed equal to the mean density of the earth. For a mass of water these numbers must be multiplied by ✓5:5; as the time of rotation to give the same figure is inversely as the square root of the density.
For å homogeneous liquid mass, of the earth's mean density, rotating in 23h46m 4s we find e=0.093, which corresponds to an ellipticity of about zo
717. An interesting form of this problem, also discussed by Laplace, is that in which the moment of momentum and the mass of the fluid are given, not the angular velocity; and it is required to find what is the eccentricity of the corresponding ellipsoid of revolution, the result proving that there can be but one.
It is evident that a mass of any ordinary liquid (not a perfect fluid, § 684), if left to itself in any state of motion, must preserve unchanged its moment of momentum, § 202. But the viscosity, or internal friction, $ 684, will, if the mass remain continuous, ultimately destroy all relative motion among its parts; so that it will ultimately rotate as a rigid solid. If the final form be an ellipsoid of revolution, we can easily show that there is a single definite value of its eccentricity. But, as it has not yet been discovered whether there is any other form consistent with stable equilibrium, we do not know that the mass will necessarily assume the form of this particular ellipsoid. Nor in fact do we know whether even the ellipsoid of rotation may not become an unstable form if the moment of momentum exceed some limit depending on the mass of the fluid. We shall return to this subject in Vol. II., as it affords an excellent example of that difficult and delicate question Kinetic Stability, § 300.
If we call a the equatorial semi-axis of the ellipsoid, e its eccentricity, and w its angular velocity of rotation, the given quantities are the mass
M=fTpa’N 1-é, and the moment of momentum
A=1 Tpwa 11–. These equations, along with (2), determine the three quantities, a, e, and w.
Eliminating a between the two just written, and expressing e as before in terms of e, we have
w2 This gives
27 (1+€?) where k is a given multiple of p. Substituting in 771 (2) we have
k=(1+c = (** tan-4-3). Now the last column of the table in $ 713 shows that the value of this function of e (which vanishes with e) continually increases with e, and becomes infinite when 6 is infinite. Hence there is always one, and only one, value of e, and therefore of e, which satisfies the conditions of the problem.
718. All the above results might without much difficulty have been obtained analytically, by the discussion of the equations; but we have preferred, for once, to show by an actual case that numerical calculation may sometimes be of very great use.
719. No one seems yet to have attempted to solve the general problem of finding all the forms of equilibrium which a mass of homogeneous incompressible fluid rotating with uniform angular velocity may assume. Unless the velocity be so small that the figure differs but little from a sphere, the problem presents difficulties of an exceedingly formidable nature. It is therefore of some importance to know that we can by a synthetical process show that another form, besides that of the ellipsoid of revolution, may be compatible with equilibrium ; viz. an ellipsoid with three unequal axes, of which the least is the axis of rotation. This curious theorem was discovered by Jacobi in 1834, and seems, simple as it is, to have been enunciated by him as a challenge to the French mathematicians? For the proof we must refer to our larger work.
1 See a Paper by Liouville, Journal de l'Ecole Polytechnique, cahier xxiii., footnote to p. 290.
(a) In the case of the Simple Pendulum, a heavy particle is suspended from a point by a light inextensible string. If we suppose it to be drawn aside from the vertical position of equilibrium and allowed to fall, it will oscillate in one plane about its lowest position. When the string has an inclination 0 to the vertical, the weight mg of the particle may be resolved into mg cos O which is balanced by the tension of the string, and mg sin 0 in the direction of the tangent to the path. If I be the length of the string, the distance (along the arc) from the position of equilibrium is 10.
Now if the angle of oscillation be small (not above 3° or 4° say), the sine and the angle are nearly equal to each other. Hence the acceleration of the motion (which is rigorously g sin ) may be written go. Hence we have a case of motion in which the acceleration is proportional to the distance from a point in the path, that is, by $ 74, Simple Harmonic Motion. The square of the angular velocity in the cor
acceleration 9 responding circular motion is
u 15 displacement=7, and the period of the harmonic motion is therefore 27 N . In the case of the pendulum, the time of an oscillation from side to side of the vertical is usually
taken-and is therefore a 1
(6) Thus the times of vibration of different pendulums are as the square roots of their lengths, for any arcs of vibration, provided only these be small.
Also the times of vibration of the same pendulum at different places are inversely as the square roots of the apparent force of gravity on a unit mass at these places.
(c) It was found experimentally by Newton that pendulums of the same length vibrate in equal times at the same place whatever be the material of which their bobs are formed. This would evidently not be the case unless the weight were in every case proportional to the amount of matter in the bob.
(d) If the simple pendulum be slightly disturbed in any way from its position of equilibrium, it will in general describe very nearly an ellipse about its lowest position as centre. This is easily seen from § 82.
(e) If the arc of vibration be considerable, the motion will not be simple harmonic, and the time of vibration will be greater than that above stated; since the acceleration being as the sine of the displacement, is in less and less ratio to the displacement as the latter is greater.
In this case, the motion for any disturbance is, for one revolution, approximately elliptic as before; but the ellipse slowly turns round the vertical, in the direction in which the bob moves.
(f) The bob may, however, be so projected as to revolve uniformly in a horizontal circle, in which case the apparatus is called a Conical Pendulum. Here we have I sin 0 for the radius of the circle, and the force in the direction of the radius is T sin 0, where T is the tension of the string. T cos 0 balances mo—and thus the force in the radius of the circle is mg tan 0. The square of the angular velocity in the circle is therefore and the time of revolution 24
I cos 0
271 1 / , where h is the height of the point of suspension above the plane of the circle. Thus all conical pendulums with the same height revolve in the same time.
(9) A rigid mass oscillating about a horizontal axis, under the action of gravity, constitutes what is called a Compound Pendulum.
When in the course of its motion the body is inclined at any angle O to the position in which it hangs, when in equilibrium, it experiences from gravity, and the resistance of the supports of its axis, a couple, which is easily seen to be equal to
gWh sin 0, where W is the mass and h the distance of its centre of gravity from the axis. This couple produces (§§ 232, 235) acceleration of angular velocity, calculated by dividing the moment of the couple by the moment of inertia of the body. Hence, if I denote the moment of inertia about the supporting axis, the angular acceleration is equal to
g W sin 0
Its motion is, therefore, identical ($ (a)) with that of the simple pendulum of length equal to 1
If a rigid body be supported about an axis, which either passes very nearly through the centre of gravity, or is at a very great distance from this point, the length of the equivalent simple pendulum will be very great: and it is clear that some particular distance for the point of support from the centre of gravity will render the length