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Re: [Phys-L] field of an ellipsoidal distribution



I haven’t been following the discussion, so I may not be helping with this comment, but I’m surprised such an equation could work for a solid. Why isn’t there any dependence on the “stiffness” of the solid (some relevant modulus)? Surely a spinning lump of play dough will bulge a lot more than a spinning lump of iron. -Carl

On Aug 12, 2015, at 4:49 PM, Donald G Polvani <dgpolvani@verizon.net> wrote:

I was interested to learn, from the posts of David Bowman and John Denker,
of the simple, but elegant, Somigliana equation for the dependence on
latitude and eccentricity of the surface gravity of an ellipsoidal object.
I was disappointed to learn that it only applies to fluid objects. As a
retired community college adjunct physics instructor (who only taught
introductory level physics), I was hoping that it would cover the case of a
rotating solid sphere of uniform density which would be a simpler case to
present to the students (although without the derivation). I was pleased to
find that the rotating, uniform density, solid sphere case can be solved
using only introductory level physics concepts and techniques, so that both
the result and derivation could be presented to introductory level students.
The equation I arrived at is:

g(theta) = g_0*sqrt(1 - A*(2 - A)*cos^2(theta)) = g_0*sqrt(1 -
B*cos^2(theta))

Where:

theta = latitude

g_0 = GM/R^2 = surface gravitational acceleration for a non-rotating,
uniform density, solid spherical object of total mass M and radius R, with G
the gravitational constant

A = maximum centrifugal acceleration/g_0 = R*w^2/g_0, with w = object's spin
rate (similar to David's post but note his "A" is 2/3 of my "A")

B = A*(2 - A) (simply defined to make the equation look more like the
Somigliana equation)

Assuming gravity is the only force present, A has a range of 0 (no rotation)
to 1 (maximum centrifugal acceleration = g_0). In this range of A, B also
goes from 0 to 1 monotonically (although B has a maximum of 1 at A = 1 and
is symmetric in A about that point).

I was hoping that the solid sphere equation would bear a close resemblance
to the Somigliana equation (with eccentricity (e) = 0). The above equation
somewhat satisfies this desire except for the -cos^2 versus +sin^2
dependence on latitude and the fact that g_0 is now the gravitational
acceleration at the pole rather than at the equator as in the Somigliana
equation. The equation does check at the equator (theta = 0) and north pole
(theta = pi/2), increases smoothly and monotonically in-between, and also
covers the non-rotational case (w = 0).

Since the maximum (physically reasonable) value of A is 1, the uniform
density and spherical object assumptions lead to a maximum possible spin
rate of sqrt((4pi/3)*G*rho) = 2.05*sqrt(G*rho) where rho is the uniform mass
density. This is to be compared with David's maximum spin rate value of
1.18*sqrt(G*rho) for a uniform density, ellipsoidal fluid object.

I compared the results of the above rotating, uniform density, solid sphere
model with some measured gravity data for the earth that I got from Young
and Freedman, "University Physics", 13th edition, Table 13.1, p 422, 2012.
For low latitudes, the model was no more than +0.05% different than the
measured data, about -0.1% different at high latitudes, with the best
agreement (about 0.03%) at mid latitudes. Since I used a mean earth radius
in the model, a possible explanation for these results could be that at low
latitudes the model overestimates g due to a mean earth radius that is too
small compared to the actual radius, underestimates g at high latitudes due
to a mean earth radius which is too large compared to the actual radius, and
gives the best results at mid-latitudes where the mean radius is the closest
to the actual radius. In this comparison, I used for the model parameters
G, M, R, and w:

G = 6.67428 x 10^-11 N*m^2/kg^2, M = 5.972 x 10^24 kg, R = 6.371 x 10^6 m, w
= 7.2921150 x 10^-5 rad/s

These parameter choices lead to A = 0.003449893, B = 0.006887884, g_0 =
9.819944 m/s^2.

Don

Dr. Donald G. Polvani
Adjunct Faculty, Physics, Retired
Anne Arundel Community College
Arnold, MD 21012

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Carl E Mungan, Assoc Prof of Physics 410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-1363
mailto:mungan@usna.edu http://usna.edu/Users/physics/mungan/