Help With Tensors: Solving a Problem

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In summary, the conversation is about finding the inertia tensor for three equal mass points located at specific coordinates, and using it to find the principle set of axes and the products of inertia with respect to those axes. The solution involves diagonalizing the matrix and finding the eigenvalues.
  • #1
atomicpedals
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I'm very clearly not understanding something, if someone could help me put my finger on what that something is. So here's what I've got:

Problem: Three equall mass points (mass m) are located at (a,a) (a,‐a) and (‐a,‐a) (all have z=0).

a) Show I = ma^2 (3ii − ij − ji + 3jj + 6kk)

I11 I12 I13
I = ma^2 I21 I22 I23
I31 I32 I33

I11 I12 0
I = ma^2 I21 I22 0
0 0 I33

3 −1 0
I= ma^2 −1 3 0
0 0 6

I = ma2 (3ii − ij − ji + 3jj + 6kk)

b) Find the principle set of axes u1, u2, and u3 in terms of I, j, and k.
Note: I’m totally sure what exactly to go for...

c) Show
I=ma2(4u1u1 +2u2u2 +6u3u3)
= ma2 (3ii − ij − ji + 3jj + 6kk)
= ma2 [Aii + Bjj + Ckk]
= ma2[4ii + 2jj+ 6kk]
=ma2(4u1u1 +2u2u2 +6u3u3)

d) Show by explicit calculation that the products of inertia with respect to the new
axes are zero.

This solution follows from part b, once I get part b hammered out this will then follow.
 
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  • #2
For part b) you want to diagonalize your matrix, i.e. find the eigenvalues. Since inertia tensors are symmetric they are diagonalizable.
 

1. What is a tensor?

A tensor is a mathematical object that describes the relationship between different sets of data. It is represented as a multi-dimensional array of numbers and is used to perform calculations and analyze data in fields such as physics, engineering, and machine learning.

2. How do I solve a problem using tensors?

The process of solving a problem using tensors involves identifying the relevant data, understanding the relationships between the data, and then using mathematical operations to manipulate the tensors and extract useful information from them. This often involves techniques such as tensor decomposition, tensor contraction, and tensor product.

3. What are some common applications of tensors?

Tensors have a wide range of applications in various fields, including image and signal processing, fluid mechanics, quantum mechanics, and data analysis. They are also used in machine learning algorithms, particularly in deep learning, for tasks such as image and speech recognition.

4. Can I use a software program to help me with tensors?

Yes, there are many software programs and libraries available that can assist with tensor calculations and analysis. Some popular options include TensorFlow, PyTorch, and NumPy. These programs offer a variety of tools and functions for creating and manipulating tensors, as well as visualizing the data.

5. How can I improve my understanding of tensors?

To improve your understanding of tensors, it is helpful to have a strong foundation in linear algebra and multivariate calculus. You can also practice by working on various problems and exercises, and reading textbooks and articles on the topic. Additionally, attending workshops or taking online courses can also deepen your understanding of tensors.

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