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Choose the correct expression to determine the angular acceleration of connecting rod.

  • A.\(-w^{2} \sin\theta \left[\frac{n^{2}-1}{(n^{2}-\sin^{2}\theta)^{3/2}}\right]\)
  • B.\(-w^{2} \sin\theta \left[\frac{n^{2}+1}{(n^{2}-\sin^{2}\theta)^{3/2}}\right]\)
  • C.\(-w^{2} \sin\theta \left[\frac{n^{2}-1}{(n^{2}-\sin^{2}\theta)^{1/2}}\right]\)
  • D.\(-w^{2} \sin\theta \left[\frac{n^{2}-1}{(n^{2}-\sin^{2}\theta)^{5/2}}\right]\)

Answer: A

The angular acceleration is the first expression, with \(n^{2} - 1\) over \((n^{2} - \sin^{2}\theta)^{3/2}\).

With crank radius r, connecting rod length l and \(n = l/r\), the connecting rod's angle \(\beta\) satisfies \(\sin\beta = \sin\theta / n\). Differentiating twice with respect to time at constant crank speed \(\omega\) gives

\[ \begin{aligned} \alpha_{c} &= -\omega^{2} \sin\theta \\ &\quad \times \frac{n^{2} - 1}{(n^{2} - \sin^{2}\theta)^{3/2}} \end{aligned} \]

The options differ only in the numerator sign or the power in the denominator. The \(3/2\) power comes from differentiating \((n^{2} - \sin^{2}\theta)^{-1/2}\), so powers of \(1/2\) or \(5/2\), or a numerator \(n^{2} + 1\), are wrong.

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