Select the Relevant Formula: The centripetal force ($F_c$) required for a body of mass $m$ moving in a circle of radius $R$ is given by $F_c = m a_c$. Since the time period ($T$) is given, it is convenient to use the angular velocity form of centripetal acceleration, $a_c = \omega^2 R$, where $\omega = \frac{2\pi}{T}$. $F_c = m \omega^2 R = m \left(\frac{2\pi}{T}\right)^2 R$ [Source 47]

Analyze the Proportionality: Since $m$ and $T$ (and thus $\omega$) are the same for both bodies, the term $m(2\pi/T)^2$ is a constant constant ($k$). $F_c = k R$ Therefore, the centripetal force is directly proportional to the radius ($F_c \propto R$).

Calculate the Ratio: $\frac{F_1}{F_2} = \frac{R_1}{R_2}$