Answer
The dimensions for all three expressions are $[L][T]^{-2}$ which are the correct dimensions for acceleration.
Work Step by Step
The units for acceleration are $m/s^2$
Therefore, the dimensions for acceleration are $[L][T]^{-2}$, where $[L]$ is length and $[T]$ is time.
The dimensions for velocity are $[L][T]^{-1}$
The dimensions for angular velocity are $[T]^{-1}$
The dimensions for radius are $[L]$
We can verify the dimensions of the three expressions for radial acceleration:
The dimensions for $v\omega$ are $[L][T]^{-1}[T]^{-1} = [L][T]^{-2}$
The dimensions for $\frac{v^2}{r}$ are $\frac{[L]^2[T]^{-2}}{[L]} = [L][T]^{-2}$
The dimensions for $\omega^2~r$ are $[T]^{-2}[L] = [L][T]^{-2}$
The dimensions for all three expressions are $[L][T]^{-2}$ which are the correct dimensions for acceleration.
