Question 11.174: A particle moves along the spiral shown; determine the magni...

Logarithmic spiral.

\begin{aligned}r & =e^{b \theta} \\\dot{r} & =\frac{d r}{d t}=b e^{b \theta} \frac{d \theta}{d t}=b e^{b \theta} \dot{\theta} \\\nu_{r} & =\dot{r}=b e^{b \theta} \dot{\theta} \quad \nu_{\theta}=r \dot{\theta}=e^{b \theta} \dot{\theta} \\\nu & =\sqrt{\nu_{r}^{2}+\nu_{\theta}^{2}}=e^{b \theta} \dot{\theta} \sqrt{b^{2}+1}\end{aligned}

\nu=e^{b \theta} \sqrt{1+b^{2}} \dot{\theta}\blacktriangleleft