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Components of Acceleration Vector of a Parametric Curve

1. Given a Curve in form of a Parametric Position Vector Function \(\vec{R}\), its Acceleration Vector (denoted by \(\vec{R\hspace{1mm}''}\) or \(\vec{A}\)) can be given as Sum of its Tangential and Normal Components as
   
   \(\vec{R\hspace{1mm}''} = \vec{A} = \vec{R_T\hspace{1mm}''} + \vec{R_N\hspace{1mm}''}\)   ...(1)
   
   where
   
   \(\vec{R_T\hspace{1mm}''} =\) Tangential Component of the Acceleration Vector of the Curve, which is in the Same Direction as the Tangent of the Curve
   
   \(\vec{R_N\hspace{1mm}''} =\) Normal Component of the Acceleration Vector of the Curve, which is in the Perpendicular to the Direction of Tangent of the Curve towards the Concave Side of the Curve
2. The formulae for Tangential and Normal Components of Acceleration Vector of a Parametric Curve can be derived as follows
   
   The Tangent/Velocity Vector \(\vec{R\hspace{1mm}'}\) of the Curve can be given as its Unit Tangent Vector \(\hat{\mathbf{T}}\) Scaled by the Magnitude of its Tangent/Velocity Vector \(|\vec{R\hspace{1mm}'}|\) as follows
   
   \(\vec{R\hspace{1mm}'} = |\vec{R\hspace{1mm}'}| \hat{\mathbf{T}}\)   ...(2)
   
   Differentiating equation (2) on Both Sides we get
   
   \({\Large \frac{d\vec{R\hspace{1mm}'}}{dt}} = {\Large \frac{d|\vec{R\hspace{1mm}'}|}{dt}} \hat{\mathbf{T}} + |\vec{R\hspace{1mm}'}| {\Large \frac{d\hat{\mathbf{T}}}{dt}} \)
   
   \(\Rightarrow \vec{R\hspace{1mm}''} = |\vec{R\hspace{1mm}'}|' \hat{\mathbf{T}} + |\vec{R\hspace{1mm}'}| {\Large \frac{\vec{R\hspace{1mm}'}\hspace{1mm}\times\hspace{1mm}(\vec{R\hspace{1mm}''}\hspace{1mm}\times\hspace{1mm}\vec{R\hspace{1mm}'})} {|\vec{R\hspace{1mm}'}|^3}}= |\vec{R\hspace{1mm}'}|' \hat{\mathbf{T}} + {\Large \frac{\vec{R\hspace{1mm}'}\hspace{1mm}\times\hspace{1mm}(\vec{R\hspace{1mm}''}\hspace{1mm}\times\hspace{1mm}\vec{R\hspace{1mm}'})} {|\vec{R\hspace{1mm}'}|^2}}\)
   
   \(\Rightarrow\vec{R\hspace{1mm}''} = |\vec{R\hspace{1mm}'}|' \hat{\mathbf{T}} + |\vec{R\hspace{1mm}'}|^2 {\Large \frac{|\vec{R\hspace{1mm}'}\hspace{1mm}\times\hspace{1mm}\vec{R\hspace{1mm}''}|} {|\vec{R\hspace{1mm}'}|^3}} {\Large \frac{\vec{R\hspace{1mm}'}\hspace{1mm}\times\hspace{1mm}(\vec{R\hspace{1mm}''}\hspace{1mm}\times\hspace{1mm}\vec{R\hspace{1mm}'})} {|\vec{R\hspace{1mm}'}\hspace{1mm}\times\hspace{1mm}(\vec{R\hspace{1mm}''}\hspace{1mm}\times\hspace{1mm}\vec{R\hspace{1mm}'})|}} \)
   
   \(\Rightarrow\vec{R\hspace{1mm}''} = |\vec{R\hspace{1mm}'}|' \hat{\mathbf{T}} + |\vec{R\hspace{1mm}'}|^2 \kappa \hat{\mathbf{N}} = {\Large \frac{\vec{R\hspace{1mm}'} \cdot \vec{R\hspace{1mm}''}}{|\vec{R\hspace{1mm}'}|}} \hat{\mathbf{T}} + {\Large \frac{|\vec{R\hspace{1mm}'} \times \vec{R\hspace{1mm}''}|}{|\vec{R\hspace{1mm}'}|}} \hat{\mathbf{N}} = \vec{R_T\hspace{1mm}''} + \vec{R_N\hspace{1mm}''}\)
   
   where
   
   \({\Large \frac{d\hat{\mathbf{T}}}{dt}} ={\Large \frac{\vec{R\hspace{1mm}'}\hspace{1mm}\times\hspace{1mm}(\vec{R\hspace{1mm}''}\hspace{1mm}\times\hspace{1mm}\vec{R\hspace{1mm}'})} {|\vec{R\hspace{1mm}'}|^3}}\)
   
   \(\kappa={\Large \frac{|\vec{R\hspace{1mm}'}\hspace{1mm}\times\hspace{1mm}\vec{R\hspace{1mm}''}|} {|\vec{R\hspace{1mm}'}|^3}}= \) Curvature of the Curve
   
   \(\hat{\mathbf{N}} = {\Large \frac{\vec{R\hspace{1mm}'}\hspace{1mm}\times\hspace{1mm}(\vec{R\hspace{1mm}''}\hspace{1mm}\times\hspace{1mm}\vec{R\hspace{1mm}'})} {|\vec{R\hspace{1mm}'}\hspace{1mm}\times\hspace{1mm}(\vec{R\hspace{1mm}''}\hspace{1mm}\times\hspace{1mm}\vec{R\hspace{1mm}'})|}}=\) Unit Principal Normal Vector of the Curve
   
   \(|\vec{R\hspace{1mm}'}|' \hat{\mathbf{T}} = {\Large \frac{\vec{R\hspace{1mm}'} \cdot \vec{R\hspace{1mm}''}}{|\vec{R\hspace{1mm}'}|}} \hat{\mathbf{T}}=\) Tangential Component of the Acceleration Vector of the Curve
   
   \(|\vec{R\hspace{1mm}'}|^2 \kappa \hat{\mathbf{N}}={\Large \frac{|\vec{R\hspace{1mm}'} \times \vec{R\hspace{1mm}''}|}{|\vec{R\hspace{1mm}'}|}} \hat{\mathbf{N}} =\) Normal Component of the Acceleration Vector of the Curve
3. The Normal Component of the Acceleration Vector of the Curve is also called Centripetal Acceleration Vector of the Curve. It is directed towards the Center of Curvature of the Curve.