Not sure what version of ChatGPT is being used on my browser at the present time, but I created a couple of GPT prompts (essentially queries) and compared and contrasted the results, and against expectations.
The first is to determine if ChatGPT could infer a wobble in a classical electromagnetic dynamics experiment. The setup is very simple and straightforward, illustrated by the schematic shown below:
As one can’t (as of yet) submit the schematic as a prompt to ChatGPT and illicit a response, we need to create a natural language prompt first:
Consider a levitating rotating magnetized sphere with north and south magnetic poles located at each endpoint of the axis of rotation. The rotation speed is 1 radian per second. An iron rod of the same length is placed in parallel with the axis of rotation, and wound with wire to form an electromagnet. At what frequency would the sphere wobble if the electromagnet was driven by a slight alternating current perturbation of 1 Herz, after the natural response subsided?
This is the response as of yesterday:
The motion of the levitating rotating magnetized sphere is governed by the interplay between the magnetic field of the sphere and the magnetic field of the iron rod, which is now an electromagnet due to the wire wound around it.
When the iron rod is magnetized by the alternating current, it generates a magnetic field that interacts with the magnetic field of the rotating sphere, causing a torque that can cause the sphere to wobble. The wobbling frequency can be determined by analyzing the equation of motion for the system.
Assuming the wobbling is small, we can linearize the equation of motion by taking the first-order Taylor series expansion. The resulting equation can be expressed as:
I * d^2θ/dt^2 + γ * dθ/dt + k * θ = M * B
where:
I is the moment of inertia of the sphere
θ is the angular displacement of the sphere from its equilibrium position
γ is the damping coefficient
k is the spring constant
M is the magnetic moment of the electromagnet
B is the magnetic field of the rotating sphere
The solution to this equation of motion is a damped harmonic oscillator, which can be expressed as:
θ(t) = A * exp(-γt/(2I)) * cos(ωt + φ)
where:
A is the amplitude of the wobbling motion
ω is the frequency of the wobbling motion
φ is the phase angle
The frequency of the wobbling motion can be found by substituting the solution into the equation of motion and solving for ω. After some algebraic manipulation, we get:
ω = sqrt((k/I) – (γ^2)/(4I^2)) – MB/(Isqrt((k/I) – (γ^2)/(4I^2)))
In this case, the sphere is rotating at a constant rate of 1 radian per second, so we can assume that the magnetic field of the sphere is also constant. Therefore, the only variable that changes due to the alternating current perturbation is the magnetic moment of the electromagnet, which oscillates at a frequency of 1 Hz.
Assuming that the natural response of the system has subsided, the wobbling motion will be dominated by the frequency of the alternating current perturbation. Therefore, the frequency of the wobbling motion will also be 1 Hz.
The response is to assert that the sphere will wobble at the same frequency as the applied forcing frequency, which is the intuitive response.
Next, change the prompt to create a similar behavior, but replacing the magnetic forces with gravitational forces.
Consider a rotating planet with massive extra weight at each pole. The rotation speed is 1 radian per Earth year. A small satellite moon orbits the planet with a significant declination. At what frequency would the planet wobble if the moon’s declination cycle was 250 days, after the natural response subsided?
This is a straightforward experiment to imagine, but difficult to perform in a lab. Submitting to ChatGPT today, this is the response:
The motion of the rotating planet with massive extra weight at each pole is governed by the conservation of angular momentum. The presence of the extra weight at the poles changes the distribution of mass of the planet, which in turn affects its moment of inertia.
Assuming that the extra weight is symmetric about the axis of rotation, we can approximate the planet as an oblate spheroid, which means that the moment of inertia of the planet is not constant but depends on the angle between the axis of rotation and the axis of symmetry of the spheroid. This angle is known as the obliquity of the planet and can change due to external perturbations such as the gravitational pull of a satellite moon.
The wobbling motion of the planet due to the gravitational pull of the satellite moon can be described by the following equation of motion:
I * d^2θ/dt^2 + γ * dθ/dt + k * θ = M * sin(ψ)
where:
I is the moment of inertia of the planet
θ is the angular displacement of the planet from its equilibrium position
γ is the damping coefficient
k is the spring constant
M is the mass of the satellite moon
ψ is the angle between the axis of rotation of the planet and the direction of the gravitational force exerted by the satellite moon
The solution to this equation of motion is a damped harmonic oscillator, which can be expressed as:
θ(t) = A * exp(-γt/(2I)) * cos(ωt + φ)
where:
A is the amplitude of the wobbling motion
ω is the frequency of the wobbling motion
φ is the phase angle
The frequency of the wobbling motion can be found by substituting the solution into the equation of motion and solving for ω. After some algebraic manipulation, we get:
ω = sqrt((k/I) – (γ^2)/(4I^2)) – Msin(ψ)/(Isqrt((k/I) – (γ^2)/(4I^2)))
In this case, the planet is rotating at a constant rate of 1 radian per Earth year, which corresponds to a period of approximately 2π × 365.25 days. The mass of the satellite moon and the gravitational force it exerts on the planet are also constant, so the only variable that changes due to the moon’s declination cycle is the angle ψ.
Assuming that the natural response of the system has subsided, the wobbling motion will be dominated by the frequency of the moon’s declination cycle, which is 1/250 Hz. Therefore, the frequency of the wobbling motion will also be 1/250 Hz.
Since the math of magnetic forcing and gravitational forcing follows the same inverse square law, the answers are similar as expected. ChatGPT gets the dimension wrong, using Hz instead of 1/(day), but other than that, this is also an intuitive result, as the forced response carries through after any natural response damps out.
Does GPT understand geophysics better than geophysicists?
I did this LLM AI experiment to find out whether ChatGPT would substantiate the model of the Chandler wobble that I had published in Mathematical Geoenergy (Wiley/AGU, 2018). The premise is that the nodal declination cycle of the Moon would force the Chandler wobble to a period of 433 days via a nonlinear interaction with the annual cycle. A detailed analysis of satellite peaks in the frequency spectrum strongly supports this model. Introducing the electromagnetic analog places it on an equal footing with a model that can be verified by a controlled lab experiment, see (A) vs (B) below.
Of course, if I had created a prompt to ChatGPT to simply inquire “What causes the Chandler wobble to cycle at 433 days?”, it would have essentially provided the Wikipedia answer, which is the hazy consensus culled from research since the wobble was discovered over 100 years ago.
Note that this “physics-free” answer provided by ChatGPT has nothing to do with the moon.
ADDED 3/26/2023
This blog post has a section on ChatGPT using MatPlotLib to create a visualization of orbiting planets.
GeoEnergy Math 
https://www.technologyreview.com/2023/03/14/1069823/gpt-4-is-bigger-and-better-chatgpt-openai/
“In theory, combining text and images could allow multimodal models to understand the world better. “It might be able to tackle traditional weak points of language models, like spatial reasoning,” says Wolf.”
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Using GPT to solve fluid dynamics issues:
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Pingback: Controlled Experiments | GeoEnergy Math
Applied to GPT-4, which is supposedly better
Prompt:
Response
In this scenario, the interaction between the rotating magnetized sphere and the electromagnet is a complex phenomenon that involves both magnetic and mechanical forces. To determine the frequency at which the sphere would wobble, we would need to consider the magnetic forces between the sphere and the electromagnet, as well as the mechanical forces acting on the sphere due to its rotation.
First, let’s consider the magnetic forces. When the alternating current in the electromagnet has a frequency of 1 Hz, it will cause the magnetic field produced by the electromagnet to oscillate with the same frequency. This oscillating magnetic field will interact with the magnetic field of the rotating sphere, creating an alternating force on the sphere.
The mechanical forces acting on the sphere due to its rotation will also play a role in the wobble frequency. The rotating sphere will experience gyroscopic effects, which will tend to resist changes in the orientation of its rotation axis. The strength of these gyroscopic forces depends on the moment of inertia and angular velocity of the sphere.
To determine the wobble frequency, we would need to combine the magnetic and mechanical forces acting on the sphere and solve for the resulting motion. This would typically involve setting up a system of equations that describe the motion of the sphere under the influence of the magnetic and gyroscopic forces, and then solving for the motion of the sphere over time.
Since the exact wobble frequency depends on the specifics of the system, such as the mass and size of the sphere, the strength of the magnetic fields, and the distance between the sphere and the electromagnet, it is difficult to provide a definitive answer without more information. However, it is likely that the wobble frequency would be influenced by the 1 Hz perturbation from the electromagnet, as well as the natural frequencies of the system that arise from the magnetic and gyroscopic forces.
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For GPT-4, notice how it changes for the wobble response
Prompt:
Response
The scenario you are describing seems to involve a planet with an uneven mass distribution due to the extra weight at the poles, and a moon with an orbit that has a significant inclination relative to the equator of the planet. This will likely cause a complex gravitational interaction that could lead to a wobble in the planet’s rotation.
However, determining the exact frequency of this wobble requires a more detailed mathematical analysis that considers various factors, such as the mass of the planet, the mass and orbital characteristics of the moon, and the distribution of the extra weight at the poles.
If you are interested in obtaining a numerical result, you would need to provide more information about the system, such as the masses involved, the specific inclinations and eccentricities of the orbits, and the nature of the extra weight at the poles. Once you have these parameters, you could use numerical simulations or mathematical models to approximate the wobble frequency.
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GPT 3.5
Consider a levitating rotating magnetized sphere with north and south magnetic poles located at each endpoint of the axis of rotation. The rotation speed is 1 radian per second. An iron rod of the same length is placed in parallel with the axis of rotation, and wound with wire to form an electromagnet. At what frequency would the sphere wobble if the electromagnet was driven by a slight alternating current perturbation of 1 Herz, after the natural response subsided?
2 / 2
The situation described involves a levitating rotating magnetized sphere with north and south magnetic poles located at each endpoint of the axis of rotation. An iron rod of the same length is placed in parallel with the axis of rotation, and wound with wire to form an electromagnet. The question asks for the frequency of the sphere’s wobbling motion if the electromagnet is driven by a slight alternating current perturbation after the natural response subsides.
To solve this problem, we need to use the concept of magnetic torque, which is the torque produced on a magnetic dipole moment when it is placed in an external magnetic field. The magnetic dipole moment of a sphere is given by:
μ = (2/3) * M * R^2
where M is the magnetic moment and R is the radius of the sphere.
Assuming that the sphere is perfectly balanced and that the axis of rotation is vertical, the magnetic torque exerted on the sphere can be calculated using the formula:
τ = μ x B
where B is the magnetic field vector and “x” denotes the cross product.
Since the electromagnet is placed in parallel with the axis of rotation, we can assume that the magnetic field produced by the electromagnet is aligned with the axis of rotation. Let’s assume that the alternating current perturbation has a sinusoidal waveform with a frequency of 1 Hz, so the magnetic field produced by the electromagnet can be written as:
B = B_0 * sin(2πt)
where B_0 is the maximum magnetic field amplitude.
The magnetic torque exerted on the sphere will therefore vary sinusoidally with time. This will cause the sphere to wobble around its axis of rotation. The frequency of this wobbling motion can be calculated using the formula:
f = (1/2π) * (τ_max / I)
where τ_max is the maximum torque exerted on the sphere, and I is the moment of inertia of the sphere.
The moment of inertia of the sphere can be approximated using the formula for a solid sphere:
I = (2/5) * M * R^2
Substituting the expression for μ and B, we get:
τ = (2/3) * M * R^2 * B_0 * sin(2πt)
τ_max = (2/3) * M * R^2 * B_0
Substituting the expressions for I and τ_max, we get:
f = (3/10π) * (M * R * B_0 / M*R^2)
Simplifying, we get:
f = (3/10π) * (B_0 / R)
Substituting the values given in the problem, we get:
f = (3/10π) * (1 / 0.5 m) = 0.191 Hz
Therefore, the sphere would wobble at a frequency of approximately 0.191 Hz if the electromagnet was driven by a slight alternating current perturbation of 1 Hz after the natural response subsides.
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GPT 3.5
Consider a rotating planet with massive extra weight at each pole. The rotation speed is 1 radian per Earth year. A small satellite moon orbits the planet with a significant declination. At what frequency would the planet wobble if the moon’s declination cycle was 250 days, after the natural response subsided?
2 / 2
The situation described involves a rotating planet with additional masses at each pole, which will cause the planet to wobble or precess. Additionally, there is a small satellite moon orbiting the planet with a significant declination. The question asks for the frequency of the planet’s wobbling motion after the natural response subsides.
To solve this problem, we need to use the concept of precession, which is the gradual rotation of the rotational axis of a spinning object around a fixed point. The precession frequency can be calculated using the formula:
f = (2π / T_p)
where f is the precession frequency, and T_p is the precession period. The precession period is related to the moment of inertia of the planet and the torque exerted by the additional masses at the poles.
Since the rotation speed of the planet is 1 radian per Earth year, the angular velocity is:
ω = 1 rad / year
The moment of inertia of the planet can be calculated using the formula:
I = 0.4MR^2
where M is the mass of the planet and R is its radius. The torque exerted by the additional masses at the poles can be calculated using the formula:
τ = MgRsin(θ)
where M is the mass of each additional weight, g is the acceleration due to gravity, R is the radius of the planet, and θ is the angle between the axis of rotation and the direction of the additional weight.
Assuming that the moon’s declination cycle is significantly smaller than the precession period, we can neglect its effect on the precession frequency. Therefore, we can use the following formula to calculate the precession period:
T_p = 2π(I / τ)
Substituting the expressions for I and τ, we get:
T_p = (5 / 3) * (R^2 / g) * (1 / ω) * (cos(θ) / M)
Assuming that the additional masses at the poles are equal, we can write:
M = (2G / c^2) * (J / R)
where G is the gravitational constant, c is the speed of light, and J is the polar moment of inertia of the planet. The polar moment of inertia can be calculated using the formula:
J = (2 / 5)MR^2
Substituting the expression for M, we get:
M = (4πG / 5c^2) * (M^2R / J)
Substituting the expression for M in the formula for T_p, we get:
T_p = (15 / 8πG) * (J / MR^2) * (cos(θ) / ω)
Finally, substituting the expression for J and simplifying, we get:
T_p = (3 / 2) * (R / g) * (cos(θ) / ω)
The precession frequency is therefore:
f = (2π / T_p) = (4πg / 3R) * (ω / cos(θ))
Substituting the values given in the problem, we get:
f = (4π * 9.81 m/s^2 / 3 * 6.37e6 m) * (1 rad/year / cos(23.44°)) = 1.390e-7 Hz
Therefore, the planet would wobble at a frequency of approximately 1.390e-7 Hz after the natural response subsided.
which is 83.2667 days, very close to 1/3 of the 250 day declination cycle, yet 250 isn’t involved in the expression so this is spurious.
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GPT-4
How (via physical aliasing) could the Moon’s nodal declination cycle of 27.212 days interact with an annual (365.25 day) impulse occurring at the solstice to nonlinearly amplify the gravitational attraction at the northern pole and thus produce the Chandler wobble with a 432.7 day period?
To understand how the Moon’s nodal declination cycle and an annual impulse at the solstice could interact to produce the Chandler wobble, we first need to establish some background knowledge.
The Chandler wobble is a small oscillation of the Earth’s axis of rotation, with a period of approximately 432.7 days. This wobble is caused by the motion of Earth’s mass distribution as it responds to external and internal forces, including gravitational interactions with the Moon and the Sun.
The Moon’s nodal declination cycle of 27.212 days refers to the time it takes for the Moon’s orbital plane to return to the same position relative to the Earth’s equatorial plane. This cycle affects the declination of the Moon, which in turn influences the gravitational forces between the Earth and the Moon.
Now, let’s consider how these two cycles could interact with an annual impulse at the solstice to produce the Chandler wobble:
The Moon’s gravitational force on Earth is not uniform. Due to the nodal declination cycle, the gravitational force varies over time, causing a periodic change in the torque acting on Earth’s equatorial bulge.
During solstices, the Earth’s axis is tilted at its maximum towards or away from the Sun. This causes an annual impulse in the form of seasonal variations in the distribution of the Earth’s mass, as the polar ice caps and the atmosphere respond to changes in the solar energy received at different latitudes.
The interaction between the Moon’s nodal declination cycle and the annual solstice impulse can lead to a nonlinear amplification of the gravitational attraction at the northern pole. This interaction can be seen as a form of physical aliasing, where the periodicities of the two cycles combine in such a way that their combined effect has a longer periodicity than either of the individual cycles.
When the nodal declination cycle and the solstice impulse combine non-linearly, they can result in a modulation of the Earth’s axial torque, which can produce a periodic wobble in Earth’s rotation axis. This wobble is the Chandler wobble, with a period of around 432.7 days.
While this explanation provides a general understanding of how the Moon’s nodal declination cycle and the annual solstice impulse could interact to produce the Chandler wobble, it is important to note that the exact mechanism and all contributing factors are still not fully understood. Earth’s dynamic system is incredibly complex, and the Chandler wobble is influenced by various other factors, such as the Earth’s internal structure and fluid motion within the Earth’s core and mantle.
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GPT-4
How (via physical aliasing) could the Moon’s nodal declination cycle of 27.212 days interact with a VERY STRONG annual (365.25 day) impulse occurring at the solstice to produce the Chandler wobble with a 432.7 day period? Based on the ellipticity of the Earth a resonant wobble would occur at a period of 305 days, so this would be a forced response within that resonant window.
The Chandler wobble is a small variation in the Earth’s axis of rotation with a period of about 432.7 days. It results from the combination of several factors, including the Earth’s non-spherical shape, distribution of mass, and external forces such as the gravitational effects of the Moon and the Sun. Your question proposes a mechanism that involves the Moon’s nodal declination cycle of 27.212 days and a strong annual impulse at the solstice to produce the Chandler wobble.
To explore this possibility, let’s first break down the key elements:
Moon’s nodal declination cycle (27.212 days): This is the period of the Moon’s orbit around the Earth, during which the Moon’s orbital plane intersects the Earth’s equatorial plane at two points called nodes. The nodal cycle can create tides, which can affect the Earth’s rotation.
Annual impulse at the solstice (365.25 days): The solstice is the point in the Earth’s orbit when the tilt of its axis is either most inclined toward (summer solstice) or away (winter solstice) from the Sun. This annual event can cause seasonal variations in the distribution of mass and temperature, which in turn can impact the Earth’s rotation.
The idea of the Moon’s nodal declination cycle interacting with a strong annual impulse to produce the Chandler wobble within the resonant window of 305 days can be approached via the concept of “beating,” a phenomenon observed when two periodic functions with slightly different frequencies interact.
In this case, the two frequencies would be those of the Moon’s nodal declination cycle and the annual impulse at the solstice. We can express these frequencies as:
f1 = 1 / 27.212 days
f2 = 1 / 365.25 days
The beat frequency, which represents the combined effect of these two frequencies, can be calculated as:
f_beat = |f1 – f2|
Now, we can find the period of the beat frequency by taking the reciprocal:
T_beat = 1 / f_beat
If T_beat were to match the period of the Chandler wobble (432.7 days), that could suggest a plausible mechanism for the interaction of the Moon’s nodal declination cycle and the annual impulse to produce the observed wobble. However, it is important to note that this is a simplified approach and might not fully capture the complexities of the Earth’s rotation and its interaction with various celestial bodies.
In reality, the Chandler wobble is likely the result of a combination of many factors, and the specific mechanism of interaction between the Moon’s nodal declination cycle and the annual impulse at the solstice may only play a part in the overall phenomenon. More sophisticated models and data would be required to validate or refute the proposed interaction in detail.
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This is what Bing explains:
The wobbling frequency of a levitating rotating magnetized sphere with north and south magnetic poles located at each endpoint of the axis of rotation can be calculated using the formula:
f = (1/2π) * (μ0 * m * B / I)
where f is the wobbling frequency, μ0 is the permeability of free space, m is the magnetic dipole moment of the sphere, B is the magnetic field strength at the location of the sphere, and I is the moment of inertia of the sphere.
The natural response subsides when the wobbling frequency equals the natural frequency of oscillation of the system. The natural frequency can be calculated using:
f0 = (1/2π) * sqrt(T/I)
where T is the restoring torque acting on the sphere.
Once you have calculated f0, you can calculate f by adding 1 Hz to f0.
I hope this helps!
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This is Bing with the hypothetical satellite moon in the main post:
The text describes a hypothetical experiment involving a rotating planet with massive extra weight at each pole, and a small satellite moon orbiting the planet with a significant declination. The experiment aims to determine the frequency at which the planet would wobble if the moon’s declination cycle was 250 days, after the natural response subsided.
The motion of the rotating planet with massive extra weight at each pole is governed by the conservation of angular momentum. The presence of the extra weight at the poles changes the distribution of mass of the planet, which in turn affects its moment of inertia.
Assuming that the extra weight is symmetric about the axis of rotation, we can approximate the planet as an oblate spheroid, which means that the moment of inertia of the planet is not constant but depends on the angle between the axis of rotation and the axis of symmetry of the spheroid. This angle is known as the obliquity of the planet and can change due to external perturbations such as the gravitational pull of a satellite moon.
The wobbling motion of the planet due to the gravitational pull of the satellite moon can be described by an equation of motion that can be solved for frequency using algebraic manipulation1.
Assuming that natural response has subsided, and that there are no other external perturbations, then wobbling motion will be dominated by frequency of moon’s declination cycle which is 1/250 Hz1.
I hope this helps! Let me know if you have any other questions.
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