Anonymous at Wed, 13 Nov 2024 02:46:07 UTC No. 16472786

>>16472756
There’s dozens of YouTube videos that will explain it for you. Stop being lazy.

Anonymous at Wed, 13 Nov 2024 04:29:45 UTC No. 16472948

>>16472756
velocity is defined as the derivative of position w.r.t. time
[math]\vec{v} \equiv \frac{d \vec{r}}{d t}[/math]
for a particles experiencing a rotation about an origin, the velocity of a particle is determined by the angular velocity vector crossed with the particle's position
[math]\vec{v} = \vec{\omega} \times \vec{r}[/math]
the momentum of a particle is defined as its mass multiplied by its velocity
[math]\vec{p} \equiv m \vec{v}[/math]
the angular momentum of a particle (w.r.t. a specific basis) is defined as the cross product of the position of a particle with its momentum
[math]\vec{L} \equiv \vec{r} \times \vec{p}[/math]
expand the definition for angular momentum using the other definitions
[math]\vec{L} = \vec{r} \times[m [\vec{\omega} \times \vec{r}]][/math]
and rewrite the cross products using the BAC-CAB vector identity
[math]\vec{A} \times [\vec{B} \times \vec{C}] = \vec{B}[\vec{A}\cdot\vec{C}] - \vec{C}[\vec{A} \cdot \vec{B}][/math]
to get
[math]\vec{L} = m [\vec{\omega} [\vec{r} \cdot \vec{r}] - \vec{r} [\vec{r} \cdot \vec{\omega}]][/math]
or, factoring out the rotation vector
[math]\vec{L} = m [r^2 - \vec{r}\vec{r}^T] \vec{\omega}[/math]
this equation can be extended to multiple particles
[math]\vec{L} = \sum_i m_i [r_i^2 - \vec{r_i}\vec{r_i}^T] \vec{\omega}[/math]
it's interesting to note that the angular momentum depends on the angular velocity vector and... what is defined to be the moment of inertia
[math]\vec{L} = \mathbf{I} \vec{\omega}[/math]
where
[math]\mathbf{I} = \sum_i m_i [r_i^2 \mathbf{1} - \vec{r_i}\vec{r_i}^T][/math]
this definition is easily extended to continuous quantities
[math]\mathbf{I} = \int_V \rho [r^2 \mathbf{1} - \vec{r}\vec{r}^T] dV[/math]
you can the MoI is a matrix (it's definitely not a scalar or a vector)
you can expand the previous equation for the moment of inertia and get what is in your pic

Anonymous at Wed, 13 Nov 2024 04:32:01 UTC No. 16472951

Trampoline.

10 quintillion(^999999999) trampolines (made of jello) perfectly nestled equally In all directions

Anonymous at Wed, 13 Nov 2024 04:32:09 UTC No. 16472952

it's a tensor because it's a linear map (in this case, an angular velocity vector is mapped to an angular momentum vector)

physicists will say a lot of mumbo jumbo about what a tensor is and how it relates to how they transform, but they are spouting confusing nonsense
in that regard, the important thing to know is that matrix coefficients represents a linear map w.r.t. chosen bases, and that when you transform bases, you have to transform the coefficients in the opposite way to make everything work the same as before

Anonymous at Wed, 13 Nov 2024 04:42:03 UTC No. 16472956

from inspection, you can see the MoI is a symmetric matrix
symmetric matrices have orthonormal eigenvectors
an eigenvector is a vector that points in a direction that is only scaled by a linear transformation
the eigenvectors of the MoI are called the principle axes
since the MoI is used to relate angular velocity to angular momentum, the eigenvectors point along directions such that the angular velocity vector and angular momentum vector directions would coincide

what is interesting is that if you describe the rotation of a rigid body over time, rotations about the principle axes with the largest and smallest eigenvector are stable, however rotation around the principle axis with the middle eigenvalue, the rotation is unstable and will cause the rigid body to rapidly start tumbling. this can be seen by trying to rotate a box or book rubber-banded closed along its different axes of symmetry; two axis support stable rotation, but the third tumbles

Anonymous at Wed, 13 Nov 2024 04:49:53 UTC No. 16472960

It's when there's a 3d matrix
t. AI "engineer"

🗑️ Anonymous at Wed, 13 Nov 2024 05:33:14 UTC No. 16472993

>>16472756
In case you get confused while looking at >>16472948, v != w x r = v_perp_to_r, but it still works out the same cuz r x v = r x v_perp_to_r. Everything else should be directly understandable

Anonymous at Wed, 13 Nov 2024 06:17:18 UTC No. 16473017

"Yes, the moment of inertia tensor depends on the mass, density distribution, and shape of the object, but not directly on the medium it’s in.

In more detail:

Mass: Greater mass generally increases the moment of inertia, as there’s more matter resisting changes in rotational motion.

Density Distribution: The moment of inertia isn’t just about total mass, but also how that mass is distributed relative to the axis of rotation. Mass farther from the axis increases the moment of inertia more significantly. For example, a hollow cylinder and a solid cylinder of the same mass and outer radius will have different moments of inertia due to the distribution of their mass.

Shape: The geometry or shape of the object defines the spatial distribution of mass, which affects the tensor. Symmetrical shapes (like spheres and cubes) have simpler moment of inertia tensors, often with zero off-diagonal terms. Irregular shapes usually have more complex tensors.

So, the moment of inertia tensor encapsulates these factors—mass, density distribution, and shape—to describe how the object resists rotational motion in different directions, but it’s intrinsic to the object itself, regardless of its environment." -ChatGPFather Who Art in Silicon

Anonymous at Wed, 13 Nov 2024 09:22:53 UTC No. 16473165

tensors are just generalizations of matrices

Anonymous at Wed, 13 Nov 2024 11:26:38 UTC No. 16473243

>>16472756
*ahem*
A tensor is something that transforms like a tensor

Anonymous at Wed, 13 Nov 2024 13:46:50 UTC No. 16473342

>>16472756
It appears to be a linear transformation.

Anonymous at Wed, 13 Nov 2024 19:46:56 UTC No. 16473761

>>16472756
The classical example of tensors is the forces applied to the faces of a cube. Forces applied perpendicular to the face of the cube are called "pressure", and forces applied parallel are called "shear."
Interpret everything involving tensors as either "pressure" or "shear" forces and you'll understand 90% of physical tensor applications.

Anonymous at Wed, 13 Nov 2024 20:04:03 UTC No. 16473787

>>16473761
this is terrible advice in so many ways

Anonymous at Wed, 13 Nov 2024 20:44:03 UTC No. 16473824

Just array of numbers, transforming things. You have three axis. But nine ways to make rotations

Anonymous at Thu, 14 Nov 2024 04:58:51 UTC No. 16474358

>>16472956
that's why if you flip toss your phone it will rotate around the long axis too.