A particle moves along the spiral shown; determine the magnitude of the velocity of the particle in terms of b, { \theta } and \dot { \theta }
Question 11.173: A particle moves along the spiral shown; determine the magni...
Hyperbolic spiral.
\begin{aligned}& r=\frac{b}{\theta} \\& \dot{r}=\frac{d r}{d t}=-\frac{b}{\theta^{2}} \frac{d \theta}{d t}=-\frac{b}{\theta^{2}} \dot{\theta} \\& \nu_{r}=\dot{r}=-\frac{b}{\theta^{2}} \dot{\theta} \quad \nu_{\theta}=r \dot{\theta}=\frac{b}{\theta} \dot{\theta} \\& \nu=\sqrt{\nu_{r}^{2}+\nu_{\theta}^{2}}=b \dot{\theta} \sqrt{\left\lgroup -\frac{1}{\theta^{2}}\right\rgroup^{2}+\left\lgroup\frac{1}{\theta}\right\rgroup^{2}} \\& =\frac{b \dot{\theta}}{\theta^{2}} \sqrt{1+\theta^{2}}\end{aligned}
\nu=\frac{b}{\theta^{2}} \sqrt{1+\theta^{2}} \dot{\theta}\blacktriangleleft
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