On 10-04-2023 12:34, John Clark wrote:
On Sun, Apr 9, 2023 at 3:11 PM smitra <smi...@zonnet.nl> wrote:
GPT: FOR A SOLID SPHERE OF UNIFORM DENSITY, WE CAN DIVIDE IT
INTO SMALL volume elements, each with mass dm. The volume element
located at a distance r from the center of the sphere has a moment
of inertia relative to the x-axis given by:
dIx = dm * y^2
_> smitra: dIz = dm * (x^2 + y^2)_
I asked GPT to take a second look at the part where it said "DIX = DM
* Y^2". I did NOT say it was wrong or it had made an error of any
kind, I just asked it to look at it, but nevertheless it knew it had
made a boo-boo. GPT said this:
"GPT: I apologize for the error in my previous response. The correct
expression for the moment of inertia of a small volume element
relative to the x-axis is:
dIx = dm * (y^2 + z^2)
This takes into account the fact that the mass element is not
necessarily located on the y-axis, but rather at a distance y from the
x-axis and at a distance z from the z-axis. Similarly, the moment of
inertia of a small volume element relative to the y-axis is:
dIy = dm * (x^2 + z^2) and the moment of inertia of a small volume
element relative to the z-axis is:
dIz = dm * (x^2 + y^2)
I hope this clarifies the issue. Thank you for bringing it to my
attention."
By the way, the version of GPT I was using was NOT the one that had
the Mathematica plug-in.
John K Clark See what's on my new list at Extropolis [1]
5vd
So, it has noticed that it used the wrong formulas and it has found the
correct ones. However, it was not able to do the calculation using
spherical symmetry. It only noted in the previous answer that the answer
is spherically symmetric because it doesn't depend on the orientation of
the axis relative to which the moment of inertia is computed.
So, it tells you the steps to compute the three dimensional integral in
spherical coordinates, it is unable to see that adding up the three
moments of inertias about the x, y and z axis yields a spherically
symmetric expression that can be computed using a one dimensional
integral over the radius.
Being unable to solve a problem in a much simpler way than the standard
way is a big red flag against the idea that GPT can do any math at all.
Saibal
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