Why Isn't I12 Zero in the Inertia Tensor for a Square Rotating about an Axis?

[pimpalicous]

Okay, so I know that the index if the inertia tensor entries stand for the moment of inertia about the second index when its rotating around the first index. For instance, the top middle entry is the moment of inertia of y when the thing is rotated around the x axis.

There's a square, and the origin of a coord system is placed in the bottom left corner. If you do out the inertia tensor, I12 is -1/4ma^2 but I13 is zero. I get why this turns out this way mathematically, but physically, why isn't I12 zero too? Its not really rotating about the y axis. What do the products of inertia really mean?

 

[LightHero]

The products of inertia are actually the off-diagonal entries in the inertia tensor, which are a measure of how the inertia tensor changes with respect to a rotation about the axes. In this case, the I12 entry is not zero because the square has an asymmetry in terms of its mass distribution. When you rotate the square around the x axis, the point masses on its left side will experience a larger moment of inertia than the point masses on its right side, resulting in a net non-zero moment of inertia about the y axis. The same applies to I13, which measures the moment of inertia of the square about the z axis when it's rotated around the x axis. The fact that it's zero indicates that the mass distribution of the square is symmetric when viewed from the x axis.

 

[astrokreng]

The products of inertia represent the distribution of mass around the axes of rotation. In your example, the value of I12 being non-zero indicates that there is a difference in the distribution of mass along the x and y axes. This can be visualized by imagining a mass distribution that is not symmetrical about the x and y axes. In this case, the moment of inertia around the x axis will be different from the moment of inertia around the y axis, resulting in a non-zero value for I12.

The physical significance of the products of inertia is that they affect the rotational motion of an object. In the case of a rotating object, the products of inertia determine how the object's angular momentum is distributed and how it responds to external torques. This is why the values of I12 and I13 are important in calculating the moment of inertia for an object.

Furthermore, the products of inertia play a role in determining the principal axes of an object, which are the axes of rotation that result in the simplest equations of motion. In your example, the non-zero value of I12 indicates that the x and y axes are not the principal axes, and therefore the equations of motion may be more complicated.

In summary, the products of inertia provide information about the distribution of mass and its effect on rotational motion. They are important in understanding and predicting the behavior of rotating objects and can also help identify the principal axes of an object.