Curvature and Components of Acceleration

A Unit Tangent Vector

A unit tangent vector is defined as

(1)   \begin{equation*}\boldsymbol{T}(t) = \frac{\boldsymbol{r}^\prime (t)}{|| \boldsymbol{r}^\prime (t) ||} .\end{equation*}

Curvature

A curvature of a smooth curve C at a point is

(2)   \begin{equation*}\kappa = || \frac{d \boldsymbol{T}}{ds} ||, \quad \quad \kappa(t) = \frac{|| \boldsymbol{T}^\prime (t) ||}{|| \boldsymbol{r}^\prime (t) ||},\end{equation*}


where s is the arc length parameter.

Tangential and Normal Components of Acceleration

The acceleration vector can be written as

(3)   \begin{equation*}\boldsymbol{a}(t) = a_N \boldsymbol{N} + a_T \boldsymbol{T} ,\end{equation*}


(4)   \begin{equation*}\boldsymbol{a}(t) = \kappa v^2 \boldsymbol{N} + \frac{dv}{dt} \boldsymbol{T},\end{equation*}


where \boldsymbol{N}(t) = \frac{d\boldsymbol{T}/dt}{|| d\boldsymbol{T}/dt ||} is the unit normal to the curve.

Formulas for Tangential and Normal Components of Acceleration, and Curvature

(5)   \begin{equation*}a_T = \frac{dv}{dt} = \frac{\boldsymbol{v} . \boldsymbol{a}}{|| \boldsymbol{v} ||} = \frac{\boldsymbol{r}^\prime (t) . \boldsymbol{r}^{\prime\prime} (t)}{|| \boldsymbol{r}^\prime (t) ||}\end{equation*}


(6)   \begin{equation*}a_N = \kappa v^2 = \frac{|| \boldsymbol{v} \times \boldsymbol{a} ||}{|| \boldsymbol{v} ||} = \frac{|| \boldsymbol{r}^\prime (t) \times \boldsymbol{r}^{\prime\prime} (t) ||}{|| \boldsymbol{r}^\prime (t) ||}\end{equation*}


(7)   \begin{equation*}\kappa(t) = \frac{|| \boldsymbol{v} \times \boldsymbol{a} ||}{|| \boldsymbol{v} ||^3} = \frac{ || \boldsymbol{r}^\prime (t) \times \boldsymbol{r}^{\prime\prime} (t) ||}{|| \boldsymbol{r}^\prime (t) ||^3}\end{equation*}

Radius of Curvature

Radius of the curvature, \rho, is defined as the reciprocal of the curvature,

(8)   \begin{equation*}\rho = \frac{1}{\kappa} .\end{equation*}

10-Minute Lecture on Curvature and Components of Acceleration

Reference

Dennis G. Zill. Advanced Engineering Mathematics, 6^{th} edition. Jones & Bartlett Learning. 2016.