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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.

Find the normal component of acceleration for a moving particle if its position function is r(t) = (4cos(2πt/3))i + (-3sin(2πt/3))j, and aN(9) is requested.

Learn how to calculate the normal component of acceleration for a particle moving along a parametric path defined by r(t) = (4cos(2πt/3))i + (-3sin(2πt/3))j. Detailed steps include finding velocity, acceleration, and using tangential and normal acceleration formulas.

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