The Inertia Tensor .... Determining Components of Angular Momentum ....

In summary, the conversation discusses difficulties in understanding the derivation of the components of Angular Momentum and the Inertia Tensor in Tensor Calculus for Physics by Dwight E. Neuenschwander. The conversation then provides equations 2.9, 2.10, and 2.11, and requests help in understanding how equation 2.11 is derived from the previous equations. The conversation concludes with a demonstration by the person seeking help, with the response simplifying the demonstration and providing a clearer explanation.
  • #1
Math Amateur
Gold Member
MHB
3,990
48
TL;DR Summary
I am having trouble following Neuenschwander's logic in deriving the components of angular momentum ... as he proceed to derive the components of the inertia tensor
I am reading Tensor Calculus for Physics by Dwight E. Neuenschwander and am having difficulties in following his logic regarding proceeding to derive the components of Angular Momentum and from there the components of the Inertia Tensor ...

On page 36 we read the following:
Neuenschwander ... Section 2,2 page 36.png


In the above text by Neuenschwander we read the following:

" ... ... ## L = \int [ ( r \bullet r ) \omega - r ( r \bullet \omega ) ] dm ## ... ... ... (2.9)

Relative to an xyz coordinate system and noting that

## \omega^i = \sum_j \omega^j \delta^{ij} ## ... ... ... (2.10)

the ith component of L may be written

## L^i =\sum_j \omega^j \ \int [ \delta^{ij} ( r \bullet r ) - x^i x^j] dm ## ... ... ... (2.11) ... ... ... "Can some please (... preferably in some detail) show me how Neuenschwander derives the expression for ## L^i ### (that is equation 2.11 from the equations above 2.11 ...Help will be much appreciated ...
Now I think it will be helpful for readers of the above post to have access to the start of Section 2.2 so i am providing this ... as follows:

Neuenschwander ... section 2.2 Page 34 .png


Neuenschwander ... Section 2,2 Page 35.png


'
Hope that helps

Peter
 
Last edited:
Physics news on Phys.org
  • #2
There is not much to be shown as it is just writing equation 2.9 on index form while keeping 2.10 in mind.
 
  • #3
Thanks for the reply Orodruin ... Given that it should be straight forward i had another shot at demonstrating the Neuenschwander Equation 2.9 leads to (the compact and rather non-intuitive IMO) equation 2.11 ... I think my demonstration is OK ... but not completely sure ... details follow ...

Peter
... 2.11 follows from 2.9 ... Page 1.png
... 2.11 follows from 2.9 ... Page  2 .png
... 2.11 follows from 2.9 ... Page  3 ... .png
Hope that is OK ...

Peter
 
  • #4
You are way overcomplicating things by writing out all the terms explicitly. The ith component of any vector ##\vec v## is given by ##v_i = \vec e_i \cdot \vec v## and therefore
$$
L_i = \vec e_i\cdot \vec L
= \int [r^2 \vec e_i\cdot\vec\omega - (\vec e_i\cdot \vec x)(\vec x\cdot \vec \omega)] dm
= \int [r^2 \omega_i - x_i (\vec x\cdot \vec \omega)] dm.
$$
Now use that ##\omega_i = \delta_{ij}\omega_j## and ##\vec x \cdot \vec \omega = x_j \omega_j## (implicit sums over j due to the summation convention) and you obtain
$$
L_i
= \int [r^2 \delta_{ij}- x_i x_j]\omega_j dm
$$
where ##\omega_j## is constant and can be moved out of the integral.
 
Last edited:
  • Like
Likes Math Amateur
  • #5
Thanks Orodruin … most helpful …

Appreciate your help …

Peter
 

1. What is the Inertia Tensor and why is it important in physics?

The Inertia Tensor is a mathematical representation of an object's mass distribution and how it resists rotational motion. It is important in physics because it helps us understand the rotational dynamics of objects and is a key component in determining the object's angular momentum.

2. How is the Inertia Tensor calculated?

The Inertia Tensor is calculated by taking the integral of the mass distribution of an object over its volume. This involves breaking the object into infinitesimal pieces and summing up the contributions from each piece.

3. What are the components of angular momentum and how are they related to the Inertia Tensor?

The components of angular momentum are angular velocity, moment of inertia, and torque. The Inertia Tensor is related to the moment of inertia, which is a measure of an object's resistance to rotational motion. The larger the moment of inertia, the more difficult it is to change the object's rotational motion.

4. Can the Inertia Tensor change for an object?

Yes, the Inertia Tensor can change for an object if its mass distribution changes. For example, if an object's shape changes or if mass is added or removed from the object, the Inertia Tensor will also change.

5. How is the Inertia Tensor used in real-world applications?

The Inertia Tensor is used in many real-world applications, such as in spacecraft design, robotics, and sports equipment. It helps engineers and designers understand how objects will behave when rotating and allows them to make informed decisions about how to optimize their designs for specific applications.

Similar threads

Replies
16
Views
2K
  • Classical Physics
Replies
5
Views
2K
  • Special and General Relativity
Replies
3
Views
1K
  • Differential Geometry
Replies
7
Views
2K
  • Introductory Physics Homework Help
10
Replies
335
Views
7K
  • Advanced Physics Homework Help
Replies
8
Views
872
  • Introductory Physics Homework Help
2
Replies
45
Views
2K
  • Introductory Physics Homework Help
Replies
10
Views
807
  • Introductory Physics Homework Help
Replies
1
Views
773
Replies
13
Views
794
Back
Top