APPENDIX.

NOTE A. LECTURE IV. p. 67.

On the change produced in the time of rotation of a mass, in consequence of its condensation, or expansion.

THE particular instance, adduced in the text for the sake of illustration, is purposely limited. The conclusion depends upon these principles.

Suppose a mass M, either rigid or not, to revolve uniformly about an axis passing through the center of gravity, with an angular velocity a.

Let m be one particle, at the perpendicular distance from the axis of rotation.

Then, if the body be projected upon the plane, which is perpendicular to the axis, and passes through the center of gravity,

x2 a 2

=

projected area described by m, in an unit of time, about the center of gravity;

:. mx2 a = 2 mass x area;

.. Emr2 a

sum of all these products.

Now a is common to the whole mass.

В в

And Σmx2 = moment of inertia of the whole mass, with respect to the axis of rotation passing through the center of gravity;

cess.

= Mk2: k being found by the usual proSee Whewell's Dynamics, Book III. Chap. 111.

Now if the same mass be either contracted or expanded, and a represent the angular velocity with which it will then revolve; and Mk2= moment of inertia : by the principle of conservation of areas,

And if the original figure of the body be similar to its figure after the change has taken place, and r, r' be radii of the respective equatoreal sections, which are considered circular;

if t and

are the times of revolution with the angular velocities a, a, respectively.

If then the Earth were expanded, without altering the relative arrangement of its particles, until its radius at the equator were equal to that of the Moon's orbit, considered circular, the time of its rotation would be about

The same conclusion will obtain, if the form of the

For instance, if a sphere of uniform density, radius r, be expanded into a spheroid of uniform density, the radius of the equatoreal circular section being r'.

Since Mk is the same for the spheroid, and for a sphere, the radius of which is equal to the equatoreal radius of the spheroid, we shall still have

If we suppose a small portion, n, to be detached from the surface of the equator of the revolving mass, M; and to continue to revolve uniformly with the angular velocity a, at the distance r; either in the manner of a planet, or as a thin ring; and that the remaining matter N is condensed into an interior mass, revolving with the angular velocity a;

let Mk2 = moment of inertia of the whole mass:

Nk2 =

.contracted mass.

Then, by the principle of conservation of areas,

For instance, suppose the Earth to be at present a sphere, or a spheroid, of uniform density; the mass of which is N. And that it had once been expanded so as to form together with the matter of the Moon, n, another sphere or spheroid, of uniform density, the radius of which at the equator, was equal to that of the Moon's orbit.

the number of days, in which the Earth would have revolved together with the Moon, upon this supposition.

If we suppose the Earth, when so expanded, to have assumed the form of a cylindrical plate, of uniform density, the radius of which was equal to that of the Moon's orbit;

We may assume such an arbitrary arrangement of the particles, as would render true the equation

Thus, assuming the present form of the Earth to be a

sphere of uniform density, the radius of which

r

=

60

N = 685, n = 10, m = 27.3,