Find the mass moment of inertia of the system of point masses in Example 11.4 about an axis from the origin through the point with coordinates (2 m, - 3 m, 4 m).

Question

Accepted Answer

From Example 11.4, the moment of inertia tensor for the system of point masses is

[latex][ I ]=\left[\begin{array}{rrr}50.56 & 20.42 & -14.94 \20.42 & 39.73 & 14.90 \-14.94 & 14.90 & 52.16\end{array} ight] \left( kg \cdot m ^2 ight)[/latex]

The vector V connecting the origin with (2 m, - 3 m, 4 m) is

[latex]V =2 \hat{ i }-3 \hat{ j }+4 \hat{ k }[/latex]

The unit vector in the direction of V is

[latex]\hat{ u }_V=\frac{ V }{\| V \|}=0.3714 \hat{ i }-0.5571 \hat{ j }+0.7428 \hat{ k }[/latex]

We may consider [latex]\hat{ u }_V[/latex] as the unit vector along the x' axis of a rotated Cartesian coordinate system. Then, from Eq. (11.50) we get

[latex]I_{V^{\prime}}=\left\lfloor\begin{array}{lll}0.3714 & -0.5571 & 0.7428\end{array} ight floor\left[\begin{array}{rrr}50.56 & 20.42 & -14.94 \20.42 & 39.73 & 14.90 \-14.94 & 14.90 & 52.16\end{array} ight]\left[\begin{array}{c}0.3714 \-0.5571 \0.7428\end{array} ight][/latex]

[latex]=\left\lfloor\begin{array}{lll}0.3714 & -0.5571 & 0.7428\end{array} ight floor\left\{\begin{array}{r}-3.695 \-3.482 \24.90\end{array} ight\}=19.06 kg \cdot m ^2[/latex]

[latex]I_{x^{\prime}}=\overbrace{\left\lfloor Q_{11} Q_{12} Q_{13} ight floor}^{\lfloor ext {Row 1 } floor}\left[\begin{array}{ccc}I_x & I_{x y} & I_{x z} \I_{y x} & I_y & I_{y z} \I_{z x} & I_{z y} & I_z\end{array} ight] \overbrace{\left\{\begin{array}{l}Q_{11} \Q_{12} \Q_{13}\end{array} ight\}}^{\lfloor ext {Row 1 } floor^T}[/latex]

[latex]I_{x^{\prime} y^{\prime}}=\overbrace{\left\lfloor Q_{21} Q_{22} Q_{23} ight floor}^{\lfloor ext {Row 2 } floor}\left[\begin{array}{ccc}I_x & I_{x y} & I_{x z} \I_{y x} & I_y & I_{y z} \I_{z x} & I_{z y} & I_z\end{array} ight] \overbrace{\left\{\begin{array}{l}Q_{31} \Q_{32} \Q_{33}\end{array} ight\}}^{\lfloor ext {Row 3 } floor^T}[/latex] (11.50)

Question 11.7: Find the mass moment of inertia of the system of point masse...

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