> I have a very old version of Mathematica. I suppose I would have to buy the new version to make this happen.
> While GPT hooked up to Mathematica may greatly improve its performance,
I would not be impressed when judging that in terms of approaching
human level cognition.
A long time ago I was teaching a statistical mechanics class, the
students had to do a certain calculation for homework that involved
doing the series expansion of 1/sinh(x) to order x. I was shocked to see
that none of the students could do it. But some did come up with the
right answer, they had shown some failed attempts and then wrote that
they used Mathematica and Mathematica says that the answer is 1/x - x/6
...
The students couldn't do it because they could not use the Taylor
expansion formula due to the singularity at x = 0. Most students had
not bothered to think about how to bypass that problem, otherwise at
least a few would have written down the correct derivation
> 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.
>> 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)
On 11-Apr-2023, at 10:49 AM, Samiya Illias <samiya...@gmail.com> 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.
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