A particle moves along the spiral shown. Knowing that \dot { \theta } is constant and denoting this constant by \varpi, determine the magnitude of the acceleration of the particle in terms of b, { \theta } and \varpi.
Question 11.173: A particle moves along the spiral shown. Knowing tha...
Hyperbolic spiral. r=\frac{b}{\theta}
\begin{aligned} &\begin{aligned} & \dot{r}=\frac{d r}{d t}=-\frac{b}{\theta^2} \frac{d \theta}{d t}=-\frac{b}{\theta^2} \dot{\theta} \\ & v_r=\dot{r}=-\frac{b}{\theta^2} \dot{\theta} \quad v_\theta=r \dot{\theta}=\frac{b}{\theta} \dot{\theta} \\ & v=\sqrt{v_r^2+v_\theta^2}=b \dot{\theta} \sqrt{\left(-\frac{1}{\theta^2}\right)^2+\left(\frac{1}{\theta}\right)^2} \\ &=\frac{b \dot{\theta}}{\theta^2} \sqrt{1+\theta^2} \end{aligned}\\ &v=\frac{b}{\theta^2} \sqrt{1+\theta^2} \dot{\theta} \end{aligned}Related Answered Questions
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