XPost: alt.magick, sci.physics.relativity

Gyroscopes are well-known for their ability to maintain stability and resist
changes in orientation. Their behavior is governed by precession, a
principle that describes how a spinning object responds to external forces.


If you drop a spinning gyroscope alongside a regular object, the gyroscope
will not simply fall straight down. It will follow a slower spiraling
path and land after the other object.

You can also usr a heavy wheel mounted on an axle, spinning rapidly in a
vertical plane. If you rotate the axle in a horizontal plane while the
wheel is still spinning, the wheel will either float upward or sink
downward, depending on the direction of rotation. This is a 90 degree
movement up or down.

You can watch that experiment here:

https://youtu.be/GeyDf4ooPdo?si=qrxh4EmBG1IhxzkD

I have used AI to create formulas for measuring the distance the gyroscope
moves in a time period while it remains still relative to the earth. There
are also two python programs. The first calculates distance and the second
makes a 3d visualization of the path of a point on the gyroscope.

The total distance traveled by a point on the wheel consists of two main
components:

Distance from the wheel's own rotation

A point on the edge of the wheel follows a circular path with a
circumference of πd.

If the wheel rotates r1 times per second, the distance covered due to the
wheel's own spin per second is: Dw=πd * r1

Distance from the axle’s rotation

The axle rotates r2 times per second, and since the wheel is attached at a
distance L from the center of the axle, the wheel follows a circular path
of radius L.

The circumference of this larger path is 2π > L2, so the distance covered
per second due to this motion is: Da=2π * L * r2

Total Distance Traveled Per Second

The total distance a point on the wheel travels in one second is the sum of
both contributions:
Dt=πd * r1+2π * L * r2

This equation gives the total linear distance a single point on the wheel
moves per second, considering both the spinning of the wheel and the
rotation around the axle.

If the wheel tilts 90 degrees upward after n full rotations of the axle, the
motion becomes more complex because the orientation of the spinning wheel
changes gradually over time. This introduces an additional tilting motion,
which affects the trajectory of a point on the wheel.

Understanding the Motion Components:

Rotation of the Wheel

The wheel itself spins at r1 rotations per second.

A point on the wheel moves a distance of πd * r1 per second due to this
spin.

Rotation Around the Axle

The wheel is attached at a distance L from the axle, creating a circular
path of radius L.
The axle rotates r2 times per second, so the distance traveled per second in
this motion is 2π * L * r2

Tilting of the Wheel

After n full rotations of the axle, the wheel tilts 90 degrees (from
horizontal to vertical).

This means the plane of the wheel gradually shifts over time, causing the
trajectory of a point on the wheel to trace a helical path in space.

Incorporating the Tilting Motion

To model this, we introduce an angular tilt rate:

The axle completes one full rotation in 1/r2 seconds.

The wheel tilts 90∘ (π/2 radians) after n full axle rotations.

The tilt rate per second is: ωt=π / (2n (1/r2)) =(π* r2) / ( 2* n)

This is the angular velocity of the wheel tilting over time.

Since the wheel is tilting, the actual distance traveled by a point follows
a helical path, rather than a simple sum of linear motions. The total
distance needs to account for the combined effect of spinning, axle
rotation, and tilt-induced displacement.

Approximate Distance Formula (Considering the Tilt)

Since the wheel tilts smoothly over time, an approximate distance formula
is:

Dt=sqrt( (π * d * r1)^2 + (2 * π * L * r2)^2 + ( (π * d) / 2n * r1)^2)

Where the third term accounts for the additional displacement caused by
tilting over time.

This equation assumes a slow, continuous tilt, and the total path becomes a
spiral with increasing complexity as the tilt progresses. If the tilt
happens in discrete steps instead of smoothly, adjustments would be needed.

Here is a python program to calculate the distance moved by the gyroscope:

# Given example values (User can provide specific ones)
d = 1 # Wheel diameter (meters)
L = 3 # Axle length (meters)
r1 = 2 # Wheel spin rate (rotations per second)
r2 = 1 # Axle rotation rate (rotations per second)
n = 5 # Number of axle rotations for 90-degree tilt

# Compute total time period
T = n / r2 # Time required for full tilt

# Compute total distance traveled
term1 = (np.pi * d * r1) ** 2
term2 = (2 * np.pi * L * r2) ** 2
term3 = ((np.pi * d / (2 * n)) * r1) ** 2

D_total = T * np.sqrt(term1 + term2 + term3)

T, D_total

Results:

Total Time Period = 5.0 seconds
Total Distance Traveled​ = 99.40 meters

These values are based on:
Wheel diameter d = 1 meter
Axle length L = 3 meters
Wheel spin rate r1 = 2 rotations per second
Axle rotation rate r2 ​= 1 rotation per second
The wheel tilting 90 degrees after n = 5 axle rotations

Here’s a 3D visualization of the path traveled by a point on the wheel as it
spins and tilts over time.

The trajectory forms a helical curve due to the combined effects of the
wheel's spin, the axle's rotation, and the gradual 90-degree tilt.

Python visualization:

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

# Define parameters
d = 1 # Wheel diameter
L = 3 # Axle length
r1 = 2 # Wheel spin rate (rotations per second)
r2 = 1 # Axle rotation rate (rotations per second)
n = 5 # Number of axle rotations for 90-degree tilt
T = 2 * n / r2 # Total time for full tilt (based on axle rotation)

# Time steps
t = np.linspace(0, T, 1000)

# Motion equations
theta_wheel = 2 * np.pi * r1 * t # Angle from wheel spinning
theta_axle = 2 * np.pi * r2 * t # Angle from axle rotation
tilt_angle = (np.pi / 2) * (t / T) # Gradual tilt from 0 to 90 degrees

# Position in 3D space
x = L * np.cos(theta_axle) + (d / 2) * np.cos(theta_wheel) *
np.cos(tilt_angle)
y = L * np.sin(theta_axle) + (d / 2) * np.sin(theta_wheel) *
np.cos(tilt_angle)
z = (d / 2) * np.sin(tilt_angle) # Vertical displacement due to tilt

# Plotting
fig = plt.figure(figsize=(8, 8))
ax = fig.add_subplot(111, projection='3d')
ax.plot(x, y, z, label="Path of a point on the wheel", color='b') ax.scatter([0], [0], [0], color='r', s=50, label="Axle center") ax.set_xlabel("X Axis")
ax.set_ylabel("Y Axis")
ax.set_zlabel("Z Axis")
ax.set_title("3D Path of a Point on the Wheel with Tilt")
ax.legend()
plt.show()

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