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