Differentiation - Connected Rates of Change
Differentiation - Connected Rates of Change
TGM Original Questions
The diagram shows a mechanism that converts rotational motion into linear motion. The electric motor drives the crankshaft (\(OP = r\) cm) to rotate, and through the transmission of the connecting rod (\(PQ = 2r\) cm), the piston (\(Q\)) makes reciprocating motion along the axis (\(OS\)). Let the distance from \(O\) to \(Q\) be \(x\) cm and angle \(POQ = \theta\).
- Find the maximum and minimum values of \(x\).
- Show that \(x = r \left( \cos \theta + \sqrt{3 + \cos^2 \theta} \right)\).
- Given that \(\theta\) is increasing at a constant rate of \(0.3\) rad/s, at the instant when \(\theta = \frac{\pi}{6}\), find the rate of change of \(x\) in terms of \(r\) and state the direction of the motion of \(Q\).
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Answer:(i) Maximum \(x=3r\); minimum \(x=r\) (ii) \(x=r\left(\cos\theta+\sqrt{3+\cos^2\theta}\right)\) (iii) \(\dfrac{\mathrm{d}x}{\mathrm{d}t}=-\dfrac{3r}{20}\left(1+\dfrac1{\sqrt5}\right)\text{ cm s}^{-1}\); \(Q\) moves towards \(O\)
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