Parallel axis theorem: Difference between revisions

Revision as of 05:28, 1 December 2021

Revision as of 22:54, 2 December 2021

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===Derivation===

{ ''Comment - This derivation is confusing because the symbols used in it do not correspond to those in the diagram on the right. Also, "perpendicular distance" is a number; it cannot be said to "lie" on the x-axis.'' ''The text could say, "both axes intersect the x-axis" or " the common perpendiculars to the the two axes are parallel to the x-axis."'' }

We may assume, without loss of generality, that in a [[Cartesian coordinate system]] the perpendicular distance between the axes lies along the ''x''-axis and that the center of mass lies at the origin. The moment of inertia relative to the ''z''-axis is

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=== Tensor generalization ===

The parallel axis theorem can be generalized to calculations involving the [[Moment of inertia#Inertia_tensorInertia tensor|inertia tensor]]. Let {{math|''I_ij''}} denote the inertia tensor of a body as calculated at the centre of mass. Then the inertia tensor {{math|''J_ij''}} as calculated relative to a new point is

:<math>J_{ij}=I_{ij} + m\left(|\mathbf{R}|^2 \delta_{ij}-R_i R_j\right),</math>