Finding the Curvature of a Circle

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*}