# Is the moment of inertia matrix a tensor?

• krabbie
In summary, the moment of inertia matrix, I, can be represented as a dyadic product according to the transformation rule for second order tensors. However, it can also be represented using the equation I_{ij} = \sum_k m_k (r_k^2 \delta_{ij} - x_{k,i}x_{k,j}), which shows that each term is a tensor under the relevant transformations.
krabbie

## Homework Statement

Is the moment of inertia matrix a tensor? Hint: the dyadic product of two vectors transforms according to the rule for second order tensors.
• $I$ is the inertia matrix
• $L$ is the angular momentum
• $\omega$ is the angular velocity

## Homework Equations

The transformation rule for a second order tensor is: $I'_{ij} = C_{ip}C_{jq}I_{pq}$. A dyadic product of two vectors $u$ and $v$ is a matrix of the form: $A_{ij} = u_iv_j$.

## The Attempt at a Solution

In a previous homework, we proved that a dyadic product transforms according to the rule for second order tensors. I would like to show that the moment of inertia matrix is a dyadic product. However, we never defined it that way in class yet, so I am unsure of how exactly this would work. We have: $L_i = I_{ij}w_j$. Now, is it ok to then write that $L_i\frac{1}{w_j} = I_{ij}$, so that $I_{ij}$ is a dyadic product of $L$ and the vector whose components are $\frac{1}{w_i}$? That feels wrong to me, because dividing out by something with a dummy variable like that doesn't seem like it should be valid, but I'm not actually sure!

You definitely can't divide by the ## \omega_j ## that way, because it is being summed over. For example, ## L_1 = I_{11}\omega_1 + I_{12}\omega_2 + I_{13}\omega_3 ##. This last equation could be divided by ## \omega_1 ##, for example, but that would not eliminate the ## \omega ##'s from the right hand side.

I am not sure whether the moment of inertia tensor can be represented simply as a dyadic product. I would think it would be easier to use something like ## I_{ij} = \sum_k m_k (r_k^2 \delta_{ij} - x_{k,i}x_{k,j}) ## and then show that each term is a tensor under the relevant transformations.

## 1. What is a moment of inertia matrix?

A moment of inertia matrix is a mathematical representation of the distribution of mass in an object. It is used to calculate the rotational inertia of an object around different axes.

## 2. Is the moment of inertia matrix a vector or a tensor?

The moment of inertia matrix is technically a second-order tensor. However, it can be represented as a vector in certain situations, such as when dealing with a planar motion.

## 3. How is the moment of inertia matrix calculated?

The moment of inertia matrix is calculated by multiplying the mass of each point in an object by the square of its distance from a specific axis of rotation. This calculation is repeated for each axis to create a 3x3 matrix.

## 4. Can the moment of inertia matrix change for an object?

Yes, the moment of inertia matrix can change for an object if its shape or mass distribution changes. For example, a spinning figure skater can change their moment of inertia by extending their arms, which changes the distribution of their mass.

## 5. What is the significance of the moment of inertia matrix in physics?

The moment of inertia matrix is an important concept in physics because it helps explain the behavior of rotating objects. It is used in equations related to angular momentum, torque, and rotational kinetic energy. It also plays a role in the stability and control of objects in motion, such as satellites and aircraft.

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