Questions Regarding the Inertia Tensor

[sams]

In Chapter 11: Dynamics of Rigid Bodies, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, pages 415-418, Section 11.3 - Inertia Tensor, I have three questions regarding the Inertia Tensor:

1.The authors made the following statement: "neither V nor ω is characteristic of the αth particle."

What do the authors mean by the above statement and how did they take V.ω outside the relation?

2. Kronecker delta function

Shouldn't the second term in the square brackets or in the parenthesis of Equation (11.9) also contain the Kronecker delta function?

3. Physical Interpretation of the diagonals and off-diagonals of the Inertia Tensor

According to the authors, the diagonal terms are called the moments of inertia and the off-diagonal terms are called the products of inertia. What are the physical interpretations of the diagonal and the off-diagonal terms? What is the difference between them?

Thank you so much for your help.

 

[DrClaude]

What do the authors mean by the above statement and how did they take V.ω outside the relation?

They mean that all particles have the same V and the same ω, so they can be taken out of the summation, due to the distributive property of the cross product.

Shouldn't the second term in the square brackets or in the parenthesis of Equation (11.9) also contain the Kronecker delta function?

No, since the second term comes from a double sum over the components of ω and x. The Kronecker delta is introduced to transform the single sum into a double sum, so that both terms can be written together as a double sum.

According to the authors, the diagonal terms are called the moments of inertia and the off-diagonal terms are called the products of inertia. What are the physical interpretations of the diagonal and the off-diagonal terms? What is the difference between them?

There is an orthogonal system of coordinates in which the tensor of inertia is diagonal. In that case, the moments of inertia obtained are aligned with the coordinate axes and they are called the principal moments of inertia. They represent the "natural" way in which the body can rotate. Any rotation can be written as a superposition of rotations around the principal moments of inertia.

 

[sams]

No, since the second term comes from a double sum over the components of ω and x. The Kronecker delta is introduced to transform the single sum into a double sum, so that both terms can be written together as a double sum.

Equation (11.8) is obtained from Equation (11.7). How did the second term come with a double sum and not the first term as well?

 

[DrClaude]

How did the second term come with a double sum and not the first term as well?

Because you are squaring a dot product. The dot product will give you a sum of three terms, and when you square it you get a product of two sums.

I suggest you write it out. It takes only a couple of lines, and it is just simple vector algebra. All will then become clear.