# Moment of Inertia tensor SETUP, not to difficult but cant figure it out

• m0nk3y
In summary, the conversation discusses setting up the integral for calculating the inertia tensor of a cylinder in cylindrical coordinates. The person was having trouble with their boundaries, but after switching to cylindrical coordinates, they were able to solve the problem. The person expresses gratitude for the help.
m0nk3y
[SOLVED] Moment of Inertia tensor SETUP, not to difficult but can't figure it out

## Homework Statement

a) For a cylinder of mass M, radius R and height h, calculate the inertia tensor about the center of mass. What are the principal axes?

## The Attempt at a Solution

I need help with setting up the integral. I placed the origin at the center of the cylinder. So and i set my boundaries as = z going from -h/2 to h/2, y going from -R to R and x going from -sqrt(R^2-y^2) to sqrt(R^2-y^2).

However when i integrate Ixx I get zero and I know I am not suppose to. Is there a better choice for my boundaries?? My x boundaries are the problem here because they give me a ugly answer.

Thanks

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Try to use cylindrical coordinates, this will simplify the problem.

so for Ixx will the integral be
r^2sin^2(theta) + h^2 r^2sin(theta)dr d(theta) dz??

thank you

Last edited:
Not shperical coordinates, cylindrical ones!

$$x=r\,\cos\phi,\,y=r\,\sin\phi \Rightarrow d\,x\,d\,y\,d\,z=r\,d\,r\,d\,\phi\,d\,z$$

thus $I_{xx}$ reads

$$I_{xx}=\int_V \rho(r,\phi,z)\,\left(r^2\,\sin^2\phi+z^2\right)\,r\,d\,r\,d\,\phi\,d\,z$$

Last edited:
got it! I feel so dumb spending so much time trying to figure it out in cartesian coord. Thank you so much!

## What is the Moment of Inertia tensor and how is it different from Moment of Inertia?

The Moment of Inertia tensor is a mathematical concept used to describe the distribution of mass within a rigid body. It is a 3x3 matrix that takes into account not only the mass of the object, but also its shape and orientation. This is different from the traditional Moment of Inertia, which only considers the mass and distance from the axis of rotation.

## How is the Moment of Inertia tensor calculated?

The Moment of Inertia tensor can be calculated by integrating the mass distribution of an object over its volume, taking into account the distance from the axis of rotation for each infinitesimal mass element. This results in a 3x3 matrix with values representing the moments of inertia in each dimension.

## What is the significance of the Moment of Inertia tensor in physics?

The Moment of Inertia tensor is an important quantity in physics as it describes the rotational inertia of a rigid body. It is used in calculations involving rotational motion and is also a crucial component in the equations of motion for objects undergoing rotational acceleration.

## How does the Moment of Inertia tensor change with different orientations of the object?

The values in the Moment of Inertia tensor will change depending on the orientation of the object with respect to the axis of rotation. This is because the distribution of mass will be different for different orientations, resulting in different moments of inertia in each dimension. The tensor can be used to calculate the moment of inertia for any orientation of the object.

## Can the Moment of Inertia tensor be used for non-rigid bodies?

No, the Moment of Inertia tensor is only applicable for rigid bodies, as it assumes that the object's shape and mass distribution remain constant during rotation. For non-rigid bodies, the moment of inertia would need to be calculated using different methods, such as integration over time for deformable objects.

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