# Moment of inertia tensor about y-axis of a cylinder.

In summary, the conversation revolves around determining the necessary ratio of height to radius for a cylinder in order for every axis to be a principal axis with the center of mass as the origin. The homework equations involve the moment of inertia tensor and the attempt at a solution involves calculating ##I_zz## using polar coordinates and then switching to cartesian coordinates. One person suggests using cylindrical coordinates and points out a mistake in the result for ##I_zz##.

## Homework Statement

What must the ratio of height to radius of a cylinder be so that every axis
is a principal axis (with the CM as the origin)?

## Homework Equations

Moment of inertia tensor.
I need I_yy = $\sum m *(x^2 + z^2)$

## The Attempt at a Solution

I calculated I_zz = MR^2 /12 . Now I change the principal axis to y-axis. I need to determine the moment of inertia tensor of it. Polar coordinates won't work. If I go for cartesian, I do not know the limits for the dx.

#### Attachments

• cylinder.png
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You can perform the integral using cylindrical coordinates ##\vec{r} = (r\cos\theta, r\sin\theta, z)##.

Also, the result for ##I_{zz}## is not correct. You should get a different factor than 12 in the denominator.

## What is the moment of inertia tensor about the y-axis of a cylinder?

The moment of inertia tensor about the y-axis of a cylinder is a mathematical representation of the distribution of mass around the y-axis of a cylinder. It takes into account the mass, shape, and orientation of the cylinder to determine its resistance to rotational motion about the y-axis.

## How is the moment of inertia tensor about the y-axis of a cylinder calculated?

The moment of inertia tensor about the y-axis of a cylinder can be calculated by using the cylinder's mass, radius, and height, along with the parallel axis theorem. The formula is Iyy = 1/12 * m * (3r^2 + h^2), where m is the mass, r is the radius, and h is the height of the cylinder.

## What are the units of the moment of inertia tensor about the y-axis of a cylinder?

The units of the moment of inertia tensor about the y-axis of a cylinder are mass times distance squared (kg*m^2). This is because the moment of inertia is a measure of the distribution of mass in an object and is dependent on the distance from the axis of rotation.

## How does the moment of inertia tensor about the y-axis differ for different types of cylinders?

The moment of inertia tensor about the y-axis will differ for different types of cylinders, such as solid, hollow, or thin-walled cylinders. This is because the distribution of mass and the distance from the axis of rotation will vary depending on the type of cylinder. For example, a hollow cylinder will have a larger moment of inertia about the y-axis compared to a solid cylinder with the same mass and dimensions.

## Why is the moment of inertia tensor about the y-axis important in physics and engineering?

The moment of inertia tensor about the y-axis is important in physics and engineering because it helps in understanding the rotational motion of objects. It is used in calculations for torque, angular acceleration, and rotational kinetic energy. It also plays a crucial role in designing structures and machines that involve rotational motion, such as wheels, gears, and turbines.

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