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The relation between Huygens-Steiner and parallel axis theorems

  1. Aug 23, 2011 #1
    Hi,

    Let [itex]I^{\mu}_{\xi}[/itex] denotes the inertia tensor of a system of N particles; each with mass [itex]m_{i}[/itex] and [itex]\vec{r_{i}}[/itex] represents the corresponding position vector with respect to an arbitrary point P. [itex]I^{\mu}_{\xi}[/itex] can be written as:

    [tex]I^{\mu}_{\xi} = \sum_{i=1}^N m_{i} ( r_{i}^{2} \delta_{\xi}^{\mu} - r_{i,\xi} r_{i}^{\mu}) [/tex]

    Moreover, [itex]\tilde{I}[/itex][itex]^{\mu}_{\xi}[/itex] is the inertia tensor in a coordinate system in which center of mass is located at the origin, that is:

    [tex]\tilde{I}^{\mu}_{\xi} = \sum_{i=1}^N m_{i} ( \tilde{r}_{i}^{2} \delta_{\xi}^{\mu} - \tilde{r}_{i,\xi} \tilde{r}_{i}^{\mu}) [/tex]

    where [itex]\vec{\tilde{r}_{i}}[/itex] represent position vector with respect to center of mass. According to Huygens-Steiner theorem, there exist a relation of this form between [itex]I^{\mu}_{\xi}[/itex] and [itex]\tilde{I}[/itex][itex]^{\mu}_{\xi}[/itex]:

    [tex]I^{\mu}_{\xi} = \tilde{I}^{\mu}_{\xi} + \sum_{i=1}^N m_{i} ( R^{2} \delta_{\xi}^{\mu} - R_{\xi} R^{\mu}) [/tex]

    where [itex]\vec{R}[/itex] is the position vector of center of mass in the original coordinate system (with point P at the origin). I tried to prove the parallel axis theorem on the basis of Huygens-Steiner theorem. I multiplied both sides of the latter equation by [itex]n_{\mu}n^{\xi}[/itex], where [itex]\vec{n}[/itex] is a unit vector that represents the direction of rotation, i.e., if [itex]\vec{\omega}[/itex] is the angular velocity vector, [itex]\vec{\omega} = \omega\ \vec{n}[/itex].

    [tex]n_{\mu}n^{\xi} I^{\mu}_{\xi} = n_{\mu}n^{\xi} \tilde{I}^{\mu}_{\xi} + \sum_{i=1}^N m_{i} ( n_{\mu}n^{\xi} R^{2} \delta_{\xi}^{\mu} - n_{\mu}n^{\xi} R_{\xi} R^{\mu}) [/tex]

    which gives:

    [tex] I = \tilde{I} + \sum_{i=1}^N m_{i} ( n_{\mu}n^{\xi} R^{2} \delta_{\xi}^{\mu} - n_{\mu}n^{\xi} R_{\xi} R^{\mu}) \ (1)[/tex]

    [itex]I[/itex] and [itex]\tilde{I}[/itex] are moment of inertia in coordinate systems with origin fixed on P and center of mass, respectively, given [itex]\vec{n}[/itex] describing direction of rotation. Also, as [itex]\vec{n}[/itex] is defined as a unit vector

    [tex]n_{\mu}n^{\xi} R^{2} \delta_{\xi}^{\mu} = R^{2}[/tex]

    We can write equation (1) as:

    [tex] I = \tilde{I} + \sum_{i=1}^N m_{i} ( R^{2} - (n_{\mu} R^{\mu})^{2}) [/tex]

    or

    [tex] I = \tilde{I} + \sum_{i=1}^N m_{i} ( R^{2} - (\vec{n} \cdot \vec{R})^{2}) [/tex]

    It seems that, it is a generalization of the parallel axis theorem to three dimensions: if our particles are distributed in a plane (hence a two dimensional distribution), and the direction of rotation is perpendicular to the resulting plane, ([itex] \vec{n} \cdot \vec{R} = 0 [/itex]) we get the following equation:

    [tex] I = \tilde{I} + \sum_{i=1}^N m_{i} R^{2} [/tex]

    which is the parallel axis theorem.

    - Is parallel axis theorem a notion which only mention to this two dimensional case? If yes, then what this three dimensional generalization is named (if it has been named at all)?

    - It seems that, the parallel axis theorem concerns relations between the moment of inertia in an arbitrary coordinate system and the center of mass frame, given a direction of rotation. On the other hand, Huygens-Steiner theorem concerns relation between inertia tensors in these two frames. Is it a correct picture?


    Here, I want to thanks for your attention and apologize if it was too long.

    Best regards and thanks again,
    Ali
     
  2. jcsd
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