Find the curvature of a circle of radius $R.$

Solution For a circle of radius $R,$ $$\begin{equation*} \mathbf{r}=\mathbf{r}( t)\;=\;R\;\cos t\mathbf{i}+R\;\sin t\mathbf{j}\qquad 0\leq t\leq 2\pi \end{equation*}$$

Then we find $\mathbf{r}^{\prime} (t)$ and $\left\Vert \mathbf{r^{\prime} }(t) \right\Vert.$ $$\begin{equation*} \begin{array}{rrr} \mathbf{r}^{\prime} ( t)\;=\;-R\;\sin t\mathbf{i}+R\;\cos t\mathbf{j}\qquad \Vert \mathbf{r}^{\prime} (t) \Vert\;=\;\sqrt{R^{2}\sin ^{2}t+R^{2}\cos ^{2}t}=R \end{array} \end{equation*}$$