体对角线为轴的转动惯量

\begin{pmatrix}

\frac{1}{6}ml^2 & 0 & 0

\\ 0 & \frac{1}{6}ml^2 & 0

\\ 0 & 0 & \frac{1}{6}ml^2

\end{pmatrix} \] ###

2. 体对角线方向的单位向量立方体的体对角线在坐标系中可以表示为从 $(-l/2, -l/2, -l/2)$ 到 $(l/2, l/2, l/2)$。对应的方向向量为： \[ \mathbf{d} = \begin{pmatrix} l \\ l \\ l \end{pmatrix} \] 将其规范化为单位向量： \[ \hat{\mathbf{d}} = \frac{1}{\sqrt{3}}\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \] ### 3. 转动惯量的计算绕任意轴的转动惯量 $ I_{\mathbf{d}} $ 可以通过惯性张量和单位向量 $\hat{\mathbf{d}}$ 计算如下： \[ I_{\mathbf{d}} = \hat{\mathbf{d}}^T \mathbf{I} \hat{\mathbf{d}} \] 将前面的矩阵和向量代入： \[ I_{\mathbf{d}} = \begin{pmatrix} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{pmatrix} \begin{pmatrix} \frac{1}{6}ml^2 & 0 & 0 \\ 0 & \frac{1}{6}ml^2 & 0 \\ 0 & 0 & \frac{1}{6}ml^2 \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \end{pmatrix} \] 执行矩阵乘法： \[ I_{\mathbf{d}} = \begin{pmatrix} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{pmatrix} \begin{pmatrix} \frac{1}{6}ml^2 \cdot \frac{1}{\sqrt{3}} \\ \frac{1}{6}ml^2 \cdot \frac{1}{\sqrt{3}} \\ \frac{1}{6}ml^2 \cdot \frac{1}{\sqrt{3}} \end{pmatrix} \] 继续计算： \[ I_{\mathbf{d}} = \frac{1}{\sqrt{3}} \cdot \frac{1}{6}ml^2 \cdot \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{3}} \cdot \frac{1}{6}ml^2 \cdot \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{3}} \cdot \frac{1}{6}ml^2 \cdot \frac{1}{\sqrt{3}} \] \[ I_{\mathbf{d}} = 3 \cdot \frac{1}{\sqrt{3}} \cdot \frac{1}{6}ml^2 \cdot \frac{1}{\sqrt{3}} \] \[ I_{\mathbf{d}} = 3 \cdot \frac{1}{3} \cdot \frac{1}{6}ml^2 \] \[ I_{\mathbf{d}} = \frac{1}{6}ml^2 \] 因此，绕立方体体对角线的转动惯量为： \[ I_{\mathbf{d}} = \frac{1}{6}ml^2 \]