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pimpalicous
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Is the parallel axis theorem always valid for inertia tensors? We have only seen examples with flat (2d) objects and was wondering if it would also be valid for 3d objects, like a h emisphere, for example. Thanks.
adriank said:I think you meant [tex]\mathbf{J} = \mathbf{I} + m(r^2\mathbf{1} - \mathbf{r}\mathbf{r}^T)[/tex]
The Parallel Axis Theorem is a physical law that states that the moment of inertia of a rigid body about an axis parallel to its center of mass is equal to the moment of inertia about an axis through its center of mass plus the product of the body's mass and the square of the distance between the two axes.
The Parallel Axis Theorem is used to calculate the moment of inertia of an object about an axis that is not through its center of mass. This is important in rotational motion, as the moment of inertia determines how difficult it is to change the rotational motion of an object.
An inertia tensor is a mathematical representation of the distribution of mass within a rigid body. It is a 3x3 matrix that can be used to calculate the moment of inertia of an object about any given axis.
The inertia tensor is used in the calculation of the moment of inertia in the Parallel Axis Theorem. The tensor contains information about the mass distribution of an object, which is necessary for calculating the moment of inertia about a specific axis.
Yes, the Parallel Axis Theorem can be applied to all objects, as long as they are rigid bodies. It is a fundamental law of physics that applies to rotational motion and is used in many practical applications, such as engineering and mechanics.