Help understanding the mathematical expression of product of Inertia

In summary, the product of inertia, represented by Ixy, is calculated by finding the moment of inertia for each axis about the center of gravity and multiplying them together. For symmetrical figures, Ix'y' will be 0. This can be found without using integration by using real values and the equation Ixy = Ix'y' + AX*Y*.
  • #1
superjose
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Homework Statement


Hello! I'm stuck and confused. I've been browsing the web, and reading books but can't seem to get it right. I've been looking for the mathematical expression for product of Inertia. I've managed to understand the 50% of it. But I'm stuck on the first part.




Homework Equations


Equation:

Ixy = Ix'y' + AX*Y*

Where A = Area of the figure
X* = The center of gravity in "x" axis
Y* = The center of gravity in "y" axis


But I can't figure what's Ix'y'. I know that if the figure is symmetrical it will give 0. But how is that possible?

Please note that I'm trying to solve problems without integration. I've been given real values.

Thanks a lot in advance.

The Attempt at a Solution

 
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  • #2
Ix'y' represents the product of inertia at a given point (x', y') relative to the center of gravity of the figure. This is calculated by taking the moment of inertia for each axis about the center of gravity and multiplying it together. For example, for a rectangle of dimensions L x W, the product of inertia would be:Ix'y' = LW3/12 - (LW2/4)*(L/2)*(W/2) where L is the length of the rectangle and W is the width. The first term is the moment of inertia of the rectangle about the x-axis and the second term is the moment of inertia of the rectangle about the y-axis. The total product of inertia can then be found by adding Ix'y' to the product of the area of the figure and the distance between the center of gravity and the point (x', y'):Ixy = Ix'y' + AX*Y* Hope this helps!
 

What is the product of inertia?

The product of inertia is a mathematical term that describes the distribution of mass around a given axis of rotation. It is also known as the second moment of mass or the moment of inertia.

How is the product of inertia calculated?

The product of inertia can be calculated by multiplying the mass of each element by the square of its distance from the axis of rotation and summing these values for all elements in the system.

What is the significance of the product of inertia?

The product of inertia is an important quantity in rotational dynamics as it determines the resistance of an object to rotational motion around a given axis. It also plays a role in determining the stability and dynamics of rotating systems.

How does the product of inertia differ from moment of inertia?

The moment of inertia is a measure of an object's resistance to rotational motion around a particular axis, while the product of inertia describes how mass is distributed around that axis. While they are related, they are not the same value.

Can the product of inertia be negative?

Yes, the product of inertia can be negative. This means that the mass is distributed unevenly around the axis of rotation, resulting in a system that is not symmetric in its rotation. However, in most cases, the product of inertia is a positive value.

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