# How Is the Moment of Inertia Tensor Calculated for a Cuboid?

• Logarythmic
In summary, the conversation is about finding the moment of inertia tensor for a cuboid of constant mass density. The solution involves using a matrix with coefficients that depend on the lengths of the cuboid's edges. The conversation also discusses the symmetry of the matrix and how it can be represented in different ways. Finally, one of the individuals requests an explanation for a specific equality involving integrals and the density function of the cuboid.
Logarythmic

## Homework Statement

Compute the moment of inertia tensor I with respect to the origin for a cuboid of constant mass density whose edges (of lengths a, b, c) are along the x,y,z-axes, with one corner at the origin.

## The Attempt at a Solution

I get

$$I = M \left( \begin{array}{ccc} \frac{1}{3} (b^2+c^2) & -\frac{ab}{4} & -\frac{ac}{4}\\ -\frac{ab}{4} & \frac{1}{3} (a^2 + c^2) & -\frac{bc}{4}\\ -\frac{ac}{4} & -\frac{bc}{4} & \frac{1}{3} (a^2 + b^2) \end{array} \right)$$

Can this be right?

I think it may be right, since the matrix is supposed to be symmetric.

Yeah, but it can be symmetric in many ways. ;)

Can someone please explain the equality

$$\int_V \rho(\vec{r}) (r^2 \delta_{jk} - x_jx_k) dV = \int_V \rho(x,y,z) \left( \begin{array}{ccc} y^2+z^2 & -xy & -xz\\ -xy & z^2+x^2 & -yz\\ -xz & -yz & x^2+y^2 \end{array} \right)dxdydz$$

for me? I think this is the most important step in my understanding for this problem.

## 1. What is moment of inertia tensor?

Moment of inertia tensor, also known as inertia tensor, is a mathematical representation of an object's mass distribution, which describes how an object's mass is distributed around its axis of rotation.

## 2. Why is moment of inertia tensor important?

Moment of inertia tensor is important because it is used to calculate an object's resistance to rotational motion and plays a crucial role in predicting an object's behavior when subjected to rotational forces.

## 3. How is moment of inertia tensor calculated?

The moment of inertia tensor is calculated by integrating the mass of each infinitesimal element of the object multiplied by the square of its distance from the axis of rotation. This integration is done for each axis of rotation, resulting in a 3x3 matrix.

## 4. What are the applications of moment of inertia tensor?

Moment of inertia tensor has various applications in physics and engineering, including predicting the motion of rotating bodies, designing rotating machinery, and analyzing the stability of structures.

## 5. Can the moment of inertia tensor be changed?

Yes, the moment of inertia tensor can be changed by altering an object's mass distribution or its axis of rotation. This can be done by changing the shape, size, or orientation of the object.

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