On 12-04-2023 13:36, John Clark wrote:
On Tue, Apr 11, 2023 at 11:06 PM smitra <smi...@zonnet.nl> wrote:
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.
I think the problem is that you and GPT have a difference of opinion
about which is simpler, Cartesian coordinates or spherical
coordinates. And there is no disputing matters of taste.
I just add this conversation with GPT
ME: If adding up the three moments of inertias about the x, y and z
axis yields a spherically symmetric expression can the moment of
inertia formula be derived by simply using a one dimensional integral
over the radius?
GPT: "Yes, that's correct. Once we have established that the moment of
inertia of a solid sphere is spherically symmetric, we can compute it
using a one-dimensional integral over the radius. Specifically, the
moment of inertia of a solid sphere of radius R and mass M about an
axis passing through its center is given by:
I = (2/5) MR^2
This formula can be derived by integrating the spherically symmetric
expression for the moment of inertia over the volume of the sphere
using spherical coordinates, which simplifies the calculation to a
one-dimensional integral over the radius. Here's how the derivation
goes:
The moment of inertia of a solid sphere of radius R and mass M about
an axis passing through its center is given by:
I = ∫∫ (r^2 sin^2θ) ρ dV
where ρ is the density and dV is the volume element in spherical
coordinates. Since the sphere has uniform density, we can write ρ = M
/ (4/3 * π * R^3).Using the fact that the integrand is spherically
symmetric, we can write:
I = ∫∫∫ (r^2 sin^2θ) ρ dV = ρ ∫∫∫ (r^2 sin^2θ) r^2 sin
θ dr dθ dφ
Substituting in the expression for ρ and using the fact that the
integral is spherically symmetric, we can simplify the integral to:
I = (3/5) MR^2 ∫∫ sin^3θ dθ dφ
The integral over θ and φ can be evaluated to give:
∫∫ sin^3θ dθ dφ = 4π/3
Substituting this into the expression for I, we get:
I = (2/5) MR^2
THEREFORE, THE MOMENT OF INERTIA OF A SOLID SPHERE ABOUT AN AXIS
PASSING THROUGH ITS CENTER CAN BE DERIVED BY SIMPLY USING A
ONE-DIMENSIONAL INTEGRAL OVER THE RADIUS, AND IS GIVEN BY (2/5) MR^2."
John K Clark See what's on my new list at Extropolis [1]
5vd
The issue is that GPT keeps on doing the derivation in the same way
where the symmetry is broken by the choice of the axis and it then ends
up with the integral over theta, phi and r and after doing the
trigonometric integral it says that there is only an integral over r
left.
But if you just write down the formula for Ix + Iy + Iz then because for
Ik you need to integrate r^2 - distance along kth axis^2, we have to
integrate 3 r^2 - (x^2 + y^2 + z^2) = 2 r^2. So, the integrand is now
spherically symmetric as a result of having added up the moment of
inertia in 3 orthogonal directions. Since these moments of inertia are
all equal due to spherical symmetry, we have that 3 times the moment of
inertia is 2 times the integral of r^2, therefore we have:
Iz = 2/3 integral over ball of radius R of r^2 dm
You can then write dm = rho 4 pi r^2 dr and only have an integral over r
from 0 to R. Even this can be simplified by dividing Iz by M, so that we
get:
Iz/M = 2/[3 V(R)] Integral over ball of radius R of r^2 dV
We can then insert V(R) = 4/3 pi R^3 and dV = 4 pi r^2 dr and integrate
over r, but we don't even need to do that. We can simply use the fact
that V(r) is some constant c times r^3 without bothering to write down
that constant explicitly:
Integral over ball of radius R of r^2 dV
= Integral over ball of radius R of d[r^2 V(r)] - Integral over ball of
radius R of 2 r V(r) dr =
R^2 V(R) - 2/5 c R^5 = 3/5 R^2 V(R)
So, we have Iz/M = 2/[3 V(R)] 3/5 R^2 V(R) = 2/5 R^2
So, in the entire derivation I only needed to integrate over the radius
r. That's the whole point of restoring spherical symmetry. Otherwise you
end up having to evaluate integrals over the angles defined relative to
the rotation axis. ChatGPT can only do the latter because in its
database there is only a derivation based on the latter logic presented.
While there are spruce's in its database that discuss spherical symmetry
for this case, it can only then talk about that, it is unable to
actually do the much simpler calculation, simply because it cannot do
any calculation at all!
And besides this problem you put ChatGPT to the test with quite a few
other problems where the standard textbook derivation is quote a bit
more complex than another method that is easily explained. Assuming that
ChatGPT understands the math and the physics of the problem, it should
have no difficulties doing the problem in the alternative, easier way.
But what you'll find is that it can only do the problem in the more
complex way that s presented on most sources.
I can write down a list of a few simple such problems that I know of
later today...
Saibal
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