# Toy Problem to Understand Products of Inertia?

• Swamp Thing
In summary, the conversation discusses the physical meaning of the off-diagonal terms, or products of inertia, in the inertia tensor. These terms indicate the need for an external torque to maintain a non-principal rotation axis, as seen in the example of a massless shaft supporting a tilted, massive disk. This relates to the concept of dynamic balancing and is represented in the inertia tensor formula from Wikipedia.
Swamp Thing
I am a bit new to the concept of the Inertia Tensor. One question that comes to mind is, what is the physical meaning of the off-diagonals i.e. the products of inertia?

and I find it quite helpful.

D H said:
Products of inertia certainly do have physical meaning. Suppose the Ixy for some object is non-zero. That means that the object cannot perform a pure x-axis rotation without some external torque. (The same also pertains to a pure y-axis rotation.)

This seems to connect with the idea of dynamic balancing, i.e. an object is dynamically balanced IFF the rotation axis lines up with a set of axes for which the inertia tensor is diagonal. (Have I got that right?)

But I'd like to see a toy problem that exemplifies this in a simple way, and in particular, I'd like to know how it ties in with the following formula:

which is from Wikipedia https://en.wikipedia.org/wiki/Moment_of_inertia#Inertia_tensor

If there are some ##m_k x_k z_k## terms that don't all cancel out, how does that call for an external torque to maintain the rotation axis?

Last edited:
Dale
Imagined a massless shaft supported on two bearings. Between the two bearings, there is mounted a massive disk, but the disk is tilted, so that the angle between the disk axis of symmetry and the shaft axis is not zero.

When the shaft is rotated, rotating bearing reaction will rotate because this is a non-principal axis rotation of the massive disk

Thanks. Your example helps to visualize a situation where you need to exert a restraining torque in order to hold the rotation axis in place.

However, it's not directly clear to me that this model will have those non-cancelling ##m_k x_k y_k## type terms -- so this model isn't enough of a "toy" for me to understand .

For every rigid body, there exists a coordinate system in which the inertia matrix is diagonal. Such a coordinate system is called a Principal Axis System. An axis of symmetry is always a principal axis.

If you start with a principal coordinate system and make a rotation to a non-principal system, the off-diagonal terms will always appear. When the body is required (forced) to rotate about a non-principal axis, the constraint forces are required.

For the example given of the massive disk obliquely mounted on a massless shaft, it is pretty obvious what the principal axes of the disk are (the shaft has no mass). The central axis defining the disk and any pair of perpendiculars in the disk are the principal axes. The required rotation axis is not one of these, so it is non-principal.

vanhees71

## 1. What is a toy problem to understand products of inertia?

A toy problem to understand products of inertia is a simplified example used to help illustrate the concept of products of inertia in physics and engineering. It typically involves a simple object, such as a cube or sphere, and asks the researcher to calculate the products of inertia for that object.

## 2. Why is understanding products of inertia important?

Products of inertia are important in understanding the rotational motion of objects. They help determine how an object will rotate around different axes and how its mass is distributed. This information is essential in designing structures and machines that need to undergo rotational motion.

## 3. How are products of inertia calculated?

Products of inertia are calculated by multiplying the mass of each point in an object by its distance from a reference point. This is done for each axis of rotation and the results are summed together to give the total products of inertia for the object.

## 4. What factors can affect the products of inertia of an object?

The shape and mass distribution of an object can greatly affect its products of inertia. Objects with irregular shapes or uneven mass distributions will have different products of inertia compared to objects with symmetrical shapes and uniform mass distribution.

## 5. How can products of inertia be used in real-world applications?

Products of inertia are commonly used in engineering and design to calculate the stability and strength of structures and machines. They are also used in robotics and aerospace engineering to determine the rotational motion of objects and how to control it.

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