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Normal Component of Acceleration Calculator

Find normal component of acceleration step by step

The calculator will find the normal component of acceleration for the object, described by the vector-valued function, at the given point, with steps shown.

Related calculators: Curvature Calculator, Tangential Component of Acceleration Calculator

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Your Input

Find the normal component of acceleration for $$$\mathbf{\vec{r}\left(t\right)} = \left\langle t, 3 t + 1, t^{2} - 5\right\rangle$$$.

Solution

Find the derivative of $$$\mathbf{\vec{r}\left(t\right)}$$$: $$$\mathbf{\vec{r}^{\prime}\left(t\right)} = \left\langle 1, 3, 2 t\right\rangle$$$ (for steps, see derivative calculator).

Find the magnitude of $$$\mathbf{\vec{r}^{\prime}\left(t\right)}$$$: $$$\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\right\rvert} = \sqrt{4 t^{2} + 10}$$$ (for steps, see magnitude calculator).

Find the derivative of $$$\mathbf{\vec{r}^{\prime}\left(t\right)}$$$: $$$\mathbf{\vec{r}^{\prime\prime}\left(t\right)} = \left\langle 0, 0, 2\right\rangle$$$ (for steps, see derivative calculator).

Find the cross product: $$$\mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)} = \left\langle 6, -2, 0\right\rangle$$$ (for steps, see cross product calculator).

Find the magnitude of $$$\mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}$$$: $$$\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}\right\rvert} = 2 \sqrt{10}$$$ (for steps, see magnitude calculator).

Finally, the normal component of acceleration is $$$a_N\left(t\right) = \frac{\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}\right\rvert}}{\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\right\rvert}} = \frac{2 \sqrt{5}}{\sqrt{2 t^{2} + 5}}.$$$

Answer

The normal component of acceleration is $$$a_N\left(t\right) = \frac{2 \sqrt{5}}{\sqrt{2 t^{2} + 5}}$$$A.