例題 2Example 2
題目 Problem
For the disc in the following figure, \(\omega_2 = 6\ \text{rad/s}\) and \(\dot\omega_2 = 2\ \text{rad/s}^2\), both relative to the platform at the instant of interest. At this instant, \(\omega_1 = 2\ \text{rad/s}\) and \(\dot\omega_1 = -3\ \text{rad/s}^2\) for the platform relative to the ground. Find the angular acceleration vector \(\dot{\vec\omega}\) for the disc relative to the ground at the instant of interest.
解答 Solution
$$\vec\omega = \vec\omega_1 + \vec\omega_2 = \omega_1\vec k + \omega_2\vec j$$
$$\begin{aligned}
\dot{\vec\omega} &= \dot\omega_1\vec k + \dot\omega_2\vec j + \omega_1\vec k\times\omega_2\vec j \\
&= \dot\omega_1\vec k + \dot\omega_2\vec j - \omega_1\omega_2\,\vec i \\
&= -3\vec k + 2\vec j - 12\vec i \ \ \text{rad/s}^2
\end{aligned}$$
- \(\dot\omega_1\vec k\)、\(\dot\omega_2\vec j\) 是大小改變的部分；\(\omega_1\vec k\times\omega_2\vec j\) 是 \(\vec\omega_2\) 被轉台帶著轉、方向改變的部分。
對照原始圖檔
