Two cars with masses m_1 and m_2 travel in circles of radii r_1 and...

The time taken to complete one circle is the time period $T$.

Given that both cars complete one circle in equal time, their time periods are equal, $T_1 = T_2$.

The angular speed is given by $\omega = \dfrac{2\pi}{T}$.

Taking the ratio of the angular speeds of the two cars:

$\dfrac{\omega_1}{\omega_2} = \dfrac{\dfrac{2\pi}{T_1}}{\dfrac{2\pi}{T_2}} = \dfrac{T_2}{T_1}$

Since $T_1 = T_2$, we get $\dfrac{\omega_1}{\omega_2} = 1$.