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is used for several kinds of rigid body rotation problems, including: (a) Fma analysis moment equation ( M. If the mass of an object is rearranged in any way that keeps every element of mass at the same distance from the axis, the moment of inertia does not change. Also defined as the capacity of a cross-section to resist bending. It must be specified with respect to a chosen axis of rotation. So you can cut the torus along the red circle, unfold it into a cylinder, and then squash it to make a flat disk of radius $R$, with the red circle as its circumference. You found the MI of this disk about an axis through its centre and parallel to the z axis. Then you used the parallel axis theorem to find the MI about the z axis, which is distance $\rho$ from the centre of the disc. Parallel Axis Theorem, Moment of Inertia Proof.

Now we will determine the value or expression for the moment of inertia of circular section about XX axis and also about YY axis IZZ IXX + I IXX D4/64. The above calculation is plausible but flawed. The parallel axis theorem is the theorem determines the moment of inertia of a rigid body about any given axis, given that moment of inertia about the parallel axis through the center of mass of an object and the perpendicular distance between the axes. The mistake is that you assumed the red disk has a uniform density. The moment of inertia of the shape is given by the equation which is the sum of all the elemental particles masses multiplied by their distance from the rotational axis squared. In the torus the amount of mass increases with radius from the z axis.

When the torus is opened and straightened to make a cylinder, the inner side must be stretched, reducing density, while the outer side has to be compressed, increasing density. So the density of the red disk increases with distance from the z axis. This also affected your use of the parallel axis theorem, because this theorem uses the distance of the centre of mass of the disk from the z axis, not the distance of the geometrical centre of the disk from the z axis.