Let's determine the dimensions for each quantity:

1. Planck's constant ($h$): From $E = h\nu$, where $E$ is energy and $\nu$ is frequency. $[h] = \frac{[E]}{[\nu]} = \frac{[ML^2T^{-2}]}{[T^{-1}]} = [ML^2T^{-1}]$.
2. Energy ($E$): $[ML^2T^{-2}]$.
3. Momentum ($p$): $p = mv$, so $[p] = [M][LT^{-1}] = [MLT^{-1}]$.
4. Angular Momentum ($L$ or $J$): $L = I\omega = mr^2 \omega$, or $L = r \times p$. $[L] = [L][MLT^{-1}] = [ML^2T^{-1}]$.
5. Power ($P$): $P = W/t$, so $[P] = \frac{[ML^2T^{-2}]}{[T]} = [ML^2T^{-3}]$. Comparing the dimensions, Planck's constant $[ML^2T^{-1}]$ has the same dimensions as Angular Momentum $[ML^2T^{-1}]$.