Parallel axis theorem: Difference between revisions

Revision as of 11:33, 18 February 2023

Revision as of 20:45, 15 April 2023

Line 28:

:<math>I = \int (x^2 + y^2) \, dm + D^2 \int dm - 2D\int x\, dm.</math>

The first term is {{math|''I''_cm}} and the second term becomes {{math|''mDMD''²}}. The integral in the final term is a multiple of the x-coordinate of the [[center of mass]]{{snd}}which is zero since the center of mass lies at the origin. So, the equation becomes:

:<math> I = I_\mathrm{cm} + mDMD^2.</math>

=== Tensor generalization ===

The parallel axis theorem can be generalized to calculations involving the [[Moment of inertia#Inertia tensor|inertia tensor]].<ref name="Abdulghany">A. R. Abdulghany, American Journal of Physics 85, 791 (2017); doi: https://dx.doi.org/10.1119/1.4994835 .</ref> Let {{math|''I_ij''}} denote the inertia tensor of a body as calculated at the centrecenter 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>

where <math>\mathbf{R}=R_1\mathbf{\hat{x}}+R_2\mathbf{\hat{y}}+R_3\mathbf{\hat{z}}\!</math> is the displacement vector from the centrecenter of mass to the new point, and {{math|δ_''ij''}} is the [[Kronecker delta]].

For diagonal elements (when {{math|''i'' {{=}} ''j''}}), displacements perpendicular to the axis of rotation results in the above simplified version of the parallel axis theorem.