On 09-04-2023 19:56, John Clark wrote:
On Sun, Apr 9, 2023 at 12:53 PM smitra <smi...@zonnet.nl> wrote:
For some simple physics or math result consider different
ways of getting to that result where one of these ways is not widely
published and is likely not in GPT's database. Take e.g. different
ways
of computing the moment of inertia of a ball of uniform density of
radius R and mass M (relative to an axis through the center). There
are
many ways to do this, but I've not seen my favorite way of doing
this on
any webpage, which is to restore spherical symmetry by adding up the
three identical moments of inertia relative to 3 orthogonal axes.
I just send this to GPT:
_Me: __Derive the formula for the moment of inertia (relative to an
axis through the center) of a ball of uniform density of radius R and
mass M by adding up the three identical moments of inertia relative to
3 orthogonal axes and restoring spherical symmetry._
_And in less than three seconds it gave me this: _
GPT: To derive the formula for the moment of inertia of a ball of
uniform density, we will start by finding the moment of inertia
relative to three orthogonal axes passing through the center of the
ball. Let's call these axes x, y, and z.
The moment of inertia of a point mass m relative to an axis at a
distance r from the mass is given by:
I = m * r^2
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
dIx = dm * (y^2 + z^2)
I'm guessing GPT found the formula for the disk in its database and used
that here
Similarly, the moment of inertia of this volume element relative to
the y-axis is:
dIy = dm * x^2
dIy = dm * (x^2 + z^2)
The moment of inertia relative to the z-axis is:
dIz = dm * r^2
dIz = dm * (x^2 + y^2)
Not r^2 but it's r^2 for a disk wth the z-axis orthogonal to the plane
of a disk.
where r is the distance from the center of the sphere to the volume
element.
We can express the differential mass element dm in terms of the
density ρ and the volume element dV:
dm = ρ * dV
For a sphere of radius R, the volume element is:
dV = 4/3 * π * r^2 * dr
Substituting dm and dV into the expressions for dIx, dIy, and dIz, and
integrating over the volume of the sphere, we get:
Ix = ∫∫∫ y^2 * ρ * (4/3 * π * r^2 * dr) dV
Iy = ∫∫∫ x^2 * ρ * (4/3 * π * r^2 * dr) dV
Iz = ∫∫∫ r^2 * ρ * (4/3 * π * r^2 * dr) dV
where the limits of integration are:
0 ≤ r ≤ R, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π
To simplify these integrals, we can use spherical coordinates, where x
= r sin θ cos φ, y = r sin θ sin φ, and z = r cos θ. The volume
element in spherical coordinates is:
dV = r^2 sin θ dr dθ dφ
Substituting this into the expressions for Ix, Iy, and Iz, we get:
Ix = 2/5 * M * R^2
Iy = 2/5 * M * R^2
Iz = 2/5 * M * R^2
Using the correct formula's that would be the results. But even then
that's not doing the problem with restrig spehrical symmetry.
where M is the mass of the sphere, given by:
M = ρ * V = 4/3 * π * R^3 * ρ
where V is the volume of the sphere.
Note that Ix = Iy, since the sphere has spherical symmetry, and the
moments of inertia relative to any two orthogonal axes in a plane
perpendicular to the third axis are equal. Therefore, the moment of
inertia of a sphere of uniform density relative to an axis through the
center is:
I = 2/5 * M * R^2
which is the desired formula.
John K Clark See what's on my new list at Extropolis [1]
grxq
So, this shows that GPT does not know what it is doing.
Saibal
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