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- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Essential_Graduate_Physics_-_Classical_Mechanics_(Likharev)/04%3A_Rigid_Body_Motion/4.02%3A_Inertia_TensorIn this case, we are left with only the second term, which describes a pure rotation of the body about its center of mass: \[\mathbf{L}=\mathbf{L}_{\mathrm{rot}} \equiv \sum m \mathbf{r} \times(\bolds...In this case, we are left with only the second term, which describes a pure rotation of the body about its center of mass: L=Lrot≡∑mr×(ω×r) Using one more vector algebra formula, the "bac minis cab" rule, 7 we may rewrite this expression as L=∑m[ωr2−r(r⋅ω)]. Let us spell out an arbitrary Cartesian co…
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/13%3A_Rigid-body_Rotation/13.04%3A_Inertia_TensorSpecifies the inertial properties of the rotating body.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/13%3A_Rigid-body_Rotation/13.07%3A_Diagonalize_the_Inertia_TensorDetermine the eigenvalues and eigenvectors for solution.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_(Fowler)/24%3A_Motion_of_a_Rigid_Body_-_the_Inertia_Tensor/24.08%3A_Principal_Axes_Form_of_Moment_of_Inertia_TensorWe already know that the transformed matrix is diagonal, so its form has to be \sum_{n} m_{n}\left(\begin{array}{ccc} x_{n 2}^{2}+x_{n 3}^{2} & 0 & 0 \\ 0 & x_{n 3}^{2}+x_{n 1}^{2} & 0 \\ 0 & 0 & x_{n...We already know that the transformed matrix is diagonal, so its form has to be \sum_{n} m_{n}\left(\begin{array}{ccc} x_{n 2}^{2}+x_{n 3}^{2} & 0 & 0 \\ 0 & x_{n 3}^{2}+x_{n 1}^{2} & 0 \\ 0 & 0 & x_{n 1}^{2}+x_{n 2}^{2} The moments of inertia, the diagonal elements, are of course all positive. Note that no one of them can exceed the sum of the other two, although it can be equal in the (idealized) case of a two-dimensional object. I_{z}=\sum_{n}\left(x_{n}^{2}+y_{n}^{2}\right)=I_{x}+I_{y}
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/02%3A_Moments_of_Inertia/2.17%3A_Solid_Body_Rotation_and_the_Inertia_TensorIt is intended that this chapter should be limited to the calculation of the moments of inertia of bodies of various shapes, and not with the huge subject of the rotational dynamics of solid bodies, w...It is intended that this chapter should be limited to the calculation of the moments of inertia of bodies of various shapes, and not with the huge subject of the rotational dynamics of solid bodies, which requires a chapter on its own. In this section I mention merely for interest two small topics involving the principal axes.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_(Fowler)/24%3A_Motion_of_a_Rigid_Body_-_the_Inertia_Tensor/24.04%3A_The_Inertia_Tensor=\sum_{n} \dfrac{1}{2} m_{n} V^{2}+\sum_{n} m_{n} \vec{V} \cdot \vec{\Omega} \times \vec{r}_{n}+\sum_{n} \dfrac{1}{2} m_{n}\left(\vec{\Omega} \times \vec{r}_{n}\right)^{2} \sum_{n} \dfrac{1}{2} m_{n}\...=\sum_{n} \dfrac{1}{2} m_{n} V^{2}+\sum_{n} m_{n} \vec{V} \cdot \vec{\Omega} \times \vec{r}_{n}+\sum_{n} \dfrac{1}{2} m_{n}\left(\vec{\Omega} \times \vec{r}_{n}\right)^{2} \sum_{n} \dfrac{1}{2} m_{n}\left(\vec{\Omega} \times \vec{r}_{n}\right)^{2}=\sum_{n} \dfrac{1}{2} m_{n}\left[\Omega^{2} r_{n}^{2}-\left(\vec{\Omega} \cdot \vec{r}_{n}\right)^{2}\right]
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/02%3A_Moments_of_Inertia/2.18%3A_Determination_of_the_Principal_AxesThe Principals Axes are the three mutually perpendicular axes in a body about which the moment of inertia is maximized.
