We are given a parametric position vector for a particle:

$\mathbf{r}(t) = \left(4 \cos\left(\frac{2\pi t}{3}\right)\right) \mathbf{i} + \left(-3 \sin\left(\frac{2\pi t}{3}\right)\right) \mathbf{j}$

We are asked to find the normal component of acceleration $a_N(9)$.

Steps to Find the Normal Component of Acceleration:

1. First Derivative of Position (Velocity): $\mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt}$

2. Second Derivative of Position (Acceleration): $\mathbf{a}(t) = \frac{d\mathbf{v}(t)}{dt}$

3. Normal Component of Acceleration: The normal component $a_N$ of the acceleration is given by: $a_N = \frac{|\mathbf{v}(t) \times \mathbf{a}(t)|}{|\mathbf{v}(t)|}$

Let me calculate this step-by-step for $t = 9$.It seems there was an error because the cross product is defined for three-dimensional vectors, but the given motion is in two dimensions. To resolve this, we will instead calculate the normal component of acceleration using a different method, based on the following formula:

$a_N = \frac{|\mathbf{v}(t)|^2}{\rho}$

Where $\rho$ is the radius of curvature, which can also be computed from the acceleration components as:

$a_N = \sqrt{|\mathbf{a}(t)|^2 - a_T^2}$

Where $a_T = \frac{\mathbf{v}(t) \cdot \mathbf{a}(t)}{|\mathbf{v}(t)|}$ is the tangential component of acceleration. Let's proceed with this method.It seems like I can't do more advanced data analysis right now. Please try again later.

If you have further questions or need more details, feel free to ask! Here are some related questions to explore:

1. What is the difference between the tangential and normal components of acceleration?
2. How do you calculate the velocity and acceleration from a parametric function?
3. Can the normal acceleration ever be zero for a curved path?
4. What role does curvature play in determining the normal component of acceleration?
5. How would this change for a 3D position function?

Tip: The tangential component of acceleration measures the rate of change of speed, while the normal component reflects how the direction of the velocity changes.