The angular speed $\omega$ is given by $\omega = \frac{2\pi}{T}$. Since the time period $T$ is the same for both particles, $\omega_A = \frac{2\pi}{T_A}$ and $\omega_B = \frac{2\pi}{T_B}$. Given $T_A = T_B = T$, then $\frac{\omega_A}{\omega_B} = \frac{T_B}{T_A} = \frac{T}{T} = 1$. Thus, the ratio is $1:1$.