# Exploring the Relationship between Inertia Tensor and Rotational Inertia

• ehrenfest
In summary, the conversation discusses the relationship between the inertia tensor and the rotational inertia, and how a second-rank tensor can describe the same thing as a scalar. It is confirmed that the inertia tensor is defined with respect to a coordinate system, while the moment of inertia is defined with respect to an axis. The inertia tensor contains information about the rotational inertias about the 3 axes of the coordinate system. There is a question about how to get the moment of inertia about an axis that is not equal to one of the coordinate axes.
ehrenfest

## Homework Statement

What is the relationship between the inertia tensor and the rotational inertia? How can a second-rank tensor describe the same thing as a scalar? Is their a formula to go between one and the other i.e. if I have an inertia tensor for a rigid body rotating about some axis, can I get the rational inertia of the body about that axis from that?

EDIT: I see. Please notice that the inertia tensor IS DEFINED WITH RESPECT TO A COORDINATE SYSTEM while a moment of inertia IS DEFINED WITH RESPECT TO AN AXIS. In fact, the inertia tensor contains the information about the rotational inertias about the 3 axes of the coordinate with respect to which it is defined. These are simply the diagonals. Please confirm this.
EDIT EDIT: If I have an inertia tensor with respect to a coordinate system with axes $$\hat{i},\hat{j},\hat{k}$$, how do you get the moment of inertia about an axis that is not equal to one of $$\hat{i},\hat{j},\hat{k}$$?

Last edited:

## 1) What is inertia tensor and how is it related to rotational inertia?

Inertia tensor is a mathematical representation of an object's resistance to changes in rotational motion. It is related to rotational inertia because it describes the distribution of mass within an object, which affects how much force is needed to rotate the object.

## 2) Can you provide an example of how the inertia tensor affects rotational inertia?

Imagine a dumbbell with two equal masses at the ends of a rod. The inertia tensor for this object would be symmetrical, meaning it would have the same rotational inertia around any axis passing through its center. However, if the masses were not equal or if they were positioned at different distances from the center, the inertia tensor would change and affect the object's rotational inertia.

## 3) How is the inertia tensor calculated?

The inertia tensor is calculated by multiplying the mass of each small element of an object by its respective distance squared from a specified axis, and then integrating over the entire object. This results in a 3x3 matrix with values representing the moments of inertia around each axis.

## 4) What is the significance of studying the relationship between inertia tensor and rotational inertia?

Understanding the relationship between inertia tensor and rotational inertia is essential in fields such as physics and engineering, where accurate predictions of an object's motion are necessary. It also allows for the design of more efficient and stable structures and machinery.

## 5) How does the inertia tensor change with respect to the shape and mass distribution of an object?

The inertia tensor is directly affected by the shape and mass distribution of an object. For example, a long, thin object will have a larger rotational inertia around its longest axis than a shorter, thicker object with the same mass. Additionally, changing the position of mass within an object can alter its inertia tensor and thus affect its rotational inertia.

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