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k This gives
2TP (1 +)? where k is a given multiple of pt. Substituting in 771 (2) we have
:-). Now the last column of the table in $ 713 shows that the value of this function of € (which vanishes with e) continually increases with €,
and becomes infinite when e 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 whịch 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.
See a Paper by Liouville, Journal de l'École 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 to the vertical, the weight mg of the particle may be resolved into mg cos 0 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 4o say), the sine and the angle are nearly equal to each other. Hence the acceleration of the motion (which is rigorously g sin 6) may be written g 0. 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 8 responding circular motion is
and the period of the
displacement ☺ harmonic motion is therefore 27 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
(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 Tsin 0, where T'is the tension of the string. T cos 0 balances mg—and thus the force in the radius of the circle is mg tan 0. The square of the angular velocity in the circle 8
I cos 0 is therefore and the time of revolution 27
8 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.
(8) 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 @ 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 sind, where W is the mass and h the distance of its centre of gravity from the axis. This couple produces (SS 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
gW sin 0
I Its motion is, therefore, identical (s (a)) with that of the simple pen
I dulum of length equal to
Wh 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 of the corresponding simple pendulum, and, therefore, the time of vibration, least possible.
To investigate these circumstances for all axes parallel to a given line, through the centre of gravity, let k be the radius of gyration round this line, we have (S 198),
I=W (k* + ho); and, therefore, if l be the length of the isochronous simple pendulum,
h+k2 (h - k) + 2hk (h - k)
= 2k +
h The second term of the last of these forms vanishes when h=k, and is positive for all other values of h. The smallest value of l is, therefore, 2k, and this, the shortest length of the isochronous simple pendulum, is realized when the axis of support is at the distance k from the centre of inertia.
To find at what distance h, from the centre of inertia the axis must be fixed to produce a pendulum isochronous with the simple pendulum, of given length l, we have the quadratic equation
h® – hl=- k”. For the solution to be possible we have seen that I must be greater than, or at least equal to, 2k. If l= 2k, the roots of this equation are equal, k being their common value. For any value of l greater than 2k, the equation has two real roots whose sum is equal to l, and product equal to k*: hence, for any distance from the centre of inertia less than k, another distance greater than k, which is a third proportional to it and k, gives the same time of vibration; and the length of the simple pendulum corresponding to either case, is equal to the sum of the distances of the two axes from the centre of inertia. This sum is equal to the distance between them if the two axes are in one plane, through the centre of inertia, and on opposite sides of this point; and, therefore, for axes thus placed, and not equidistant from the centre of inertia, if the times of oscillation of the body when successively supported upon them are found to be equal, it may be inferred that the distance between them is equal to the length of the isochronous simple pendulum. As a simple pendulum exists only in theory, this proposition was taken advantage of by Kater for the practical determination of the force of gravity at any station.
(h) A uniformly heavy and perfectly flexible còrd, placed in the interior of a smooth tube in the form of any plane curve, and subject to no external forces, will exert no pressure on the tube if it have everywhere the same tension, and move with a certain definite velocity. For, as in $ 592, the statical pressure due to the curvature of the
0 rope per unit of length is T – (where o is the length of the arc AB in that figure) directed inwards to the centre of curvature. Now, the element
whose mass is mo, is moving in a curve whose curvature is A
mvo0 with velocity v (suppose). The requisite force is = mv'e;
and for unit of length mve- Hence if T=mvi the theorem is true. If we suppose a portion of the tube to be straight, and the whole to be moving with velocity v parallel to this line, and against the motion of the cord, we shall have the straight part of the cord reduced to rest, and an undulation, of any, but unvarying, form and dimensions,
running along it with the linear velocity VI
Suppose the cord stretched by an appended mass of W pounds, and suppose its length I feet and its own mass w pounds. Then I'= Wg, Im=w, and the velocity of the undulation is
feet per second.
() When an incompressible liquid escapes from an orifice, the velocity is the same as would be acquired by falling from the free surface to the level of the orifice.
For, as we may neglect (provided the vessel is large compared with the orifice) the kinetic energy of the bulk of the liquid; the kinetic energy of the escaping liquid is due to the loss of potential energy of the whole by the depression of the free surface. Thus the proposition at once.
(k) The small oscillations of a liquid in a U tube follow the harmonic law.
The tube being of uniform section S, a depression of level, x, from the mean, on one side, leads to a rise, x, on the other; and if the whole column of fluid be of length 2a, we have the mass za Sp disturbed through a space x, and acted on by a force 2 Sxgp tending to bring it back. The time of oscillation is therefore ( (a)) 27 and is the same for all liquids whatever be their densities.