例題 5Example 5
題目 Problem
The wheel of radius \(r\) is free to rotate about the bent axle CO which turns about the vertical axis at the constant rate \(p\ \text{rad/s}\). If the wheel rolls without slipping on the horizontal circle of radius \(R\), determine the expressions for the angular velocity \(\omega\) and angular acceleration \(\alpha\) of the wheel. The \(x\)-axis is always horizontal.
解答 Solution
$$\vec\omega = p\,\vec e_p + \frac{pR}{r}\vec k \qquad \vec e_p = \cos\theta\,\vec j + \sin\theta\,\vec k$$
$$\begin{aligned}
\vec\omega &= p\cos\theta\,\vec j + p\sin\theta\,\vec k + \frac{pR}{r}\vec k \\
&= p\cos\theta\,\vec j + p\left(\sin\theta + \frac{R}{r}\right)\vec k
\end{aligned}$$
$$\begin{aligned}
\vec\alpha = \dot{\vec\omega} &= p\,\vec e_p\times\frac{pR}{r}\vec k \\
&= \frac{p^2R}{r}\left(\cos\theta\,\vec j + \sin\theta\,\vec k\right)\times\vec k \\
&= \frac{p^2R}{r}\cos\theta\,\vec i
\end{aligned}$$
- \(\vec e_p\) 是鉛直方向（彎軸轉動軸）的單位向量；\(\frac{pR}{r}\vec k\) 是輪子對彎軸的自轉（由純滾動條件得出）。
- 編註：原圖第 1–3 行把 \(\vec e_p\) 寫成 \(\cos\theta\,\vec i + \sin\theta\,\vec k\)，倒數第 2 行又寫成 \(\cos\theta\,\vec j + \sin\theta\,\vec k\)。題目說 \(x\) 軸恆水平，鉛直單位向量不會有 \(\vec i\) 分量，且最後答案 \(\frac{p^2R}{r}\cos\theta\,\vec i\) 也只有用 \(\vec j\) 才算得出來，因此這裡統一更正為 \(\vec j\)。原圖第 4 行寫成 \(\frac{pR}{r}p\,\vec e_p\times\vec k\)，與上式相同。
對照原始圖檔
