Parallel axis theorem: Difference between revisions

Revision as of 20:45, 15 April 2023

Revision as of 04:19, 27 April 2023

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{{Short description|Theorem in planar dynamics}}

{{Redirect-distinguish|Steiner's theorem|Steiner's theorem (geometry)}}

The '''parallel axis theorem''', also known as '''Huygens&ndash;Steiner theorem''', or just as '''Steiner's theorem''',<ref>{{cite bookcitation | title=Introduction to theoretical physics | author=Arthur Erich Haas | year=1928}}</ref> named after [[Christiaan Huygens]] and [[Jakob Steiner]], can be used to determine the [[moment of inertia]] or the [[second moment of area]] of a [[rigid body]] about any axis, given the body's moment of inertia about a [[Parallel (geometry)|parallel]] axis through the object's [[center of gravity]] and the [[perpendicular]] [[distance]] between the axes.

==Mass moment of inertia==

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=== Tensor generalization ===

The parallel axis theorem can be generalized to calculations involving the [[Moment of inertia#Inertia tensor|inertia tensor]].<ref name="Abdulghany">A.{{citation
| R.last = Abdulghany, American| Journalfirst of= PhysicsA. 85,R.
| 791date = October (2017);
| doi: https://dx.doi.org/= 10.1119/1.4994835
| .issue = 10
| journal = American Journal of Physics
| pages = 791–795
| title = Generalization of parallel axis theorem for rotational inertia
| volume = 85}}</ref> Let {{math|''I_ij''}} denote the inertia tensor of a body as calculated at the center 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>

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where '''E'''₃ is the {{nobr|3 × 3}} [[identity matrix]] and <math>\otimes</math> is the [[outer product]].

Further generalization of the parallel axis theorem gives the inertia tensor about any set of orthogonal axes parallel to the reference set of axes x, y and z, associated with the reference inertia tensor, whether or not they pass through the center of mass.<ref name="Abdulghany">A. R. Abdulghany, American Journal of Physics 85, 791 (2017); doi: https://dx.doi.org/10.1119/1.4994835 .</ref>

==Second moment of area==

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==Moment of inertia matrix==

The inertia matrix of a rigid system of particles depends on the choice of the reference point.<ref name="Kane">{{citation|first1=T. R. |last1=Kane and |first2=D. A. |last2=Levinson, [https://www.amazon.com/Dynamics-Theory-Applications-Mechanical-Engineering/dp/0070378460 |title=Dynamics, Theory and Applications],|publisher= McGraw-Hill, NY,New York|year= 2005.}}</ref> There is a useful relationship between the inertia matrix relative to the center of mass '''R''' and the inertia matrix relative to another point '''S'''. This relationship is called the parallel axis theorem.

Consider the inertia matrix [I_S] obtained for a rigid system of particles measured relative to a reference point '''S''', given by