# 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

$\langle$
,
,
$\rangle$
Leave empty if you don't need the normal component at a specific point.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

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 $a_N\left(t\right) = \frac{\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}\right\rvert}}{\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\right\rvert}} = \frac{2 \sqrt{5}}{\sqrt{2 t^{2} + 5}}.$

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