From expressions above, form the acceleration vector a = arˆ r+aθ ˆ θ. Say something interesting about it.

To express the acceleration of point \( A \) in vector form, we will use polar coordinates with two components: radial and tangential.

### 1. **Radial acceleration component \( a_r \)**:
Radial acceleration is directed towards the center of the circular path of motion. Since point \( A \) is moving along a circle of radius \( R + r \), the radial component of the acceleration is given by:

\[
a_r = \frac{(R + r) \Omega^2}{R}
\]

This is the centripetal acceleration, pointing towards the center \( O \) of the ring.

### 2. **Tangential acceleration component \( a_\theta \)**:
As mentioned earlier, since the angular velocity \( \Omega \) is constant (i.e., there is no angular acceleration), there is no tangential acceleration:

\[
a_\theta = 0
\]

### 3. **Acceleration vector**:
The total acceleration \( \mathbf{a} \) of point \( A \) can therefore be written in vector form as:

\[
\mathbf{a} = a_r \hat{r} + a_\theta \hat{\theta}
\]

Substituting the values of \( a_r \) and \( a_\theta \), we get:

\[
\mathbf{a} = \frac{(R + r) \Omega^2}{R} \hat{r} + 0 \hat{\theta}
\]

This simplifies to:

\[
\mathbf{a} = \frac{(R + r

Explore how to express the acceleration vector using polar coordinates, with a breakdown of radial and tangential components. This problem explains centripetal acceleration and its application in circular motion. Suitable for high school students (Grades 11-12).

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