The centripetal acceleration, $a$, of an object in uniform circular motion is expressed in terms of angular velocity, $\omega$, and radius, $r$, as:

$$a = \omega^2 r$$

The angular velocity is related to the frequency of revolution, $f$, by $\omega = 2\pi f$. The frequency is the number of revolutions, $N$, completed in a time interval, $t$, so $f = N/t$.

First, we determine the ratio of the angular velocities. Since the time interval $t$ is the same for both balls:

$$ \frac{\omega_A}{\omega_B} = \frac{2\pi f_A}{2\pi f_B} = \frac{f_A}{f_B} = \frac{N_A/t}{N_B/t} = \frac{N_A}{N_B} $$

Next, we derive the ratio of the centripetal accelerations, $a_A:a_B$:

$$ \frac{a_A}{a_B} = \frac{\omega_A^2 r_A}{\omega_B^2 r_B} = \left(\frac{\omega_A}{\omega_B}\right)^2 \left(\frac{r_A}{r_B}\right) $$

Substituting the expression for the ratio of angular velocities:

$$ \frac{a_A}{a_B} = \left(\frac{N_A}{N_B}\right)^2 \left(\frac{r_A}{r_B}\right) $$

We are given the following information:

• Ratio of radii: $\frac{r_A}{r_B} = \frac{1}{2}$
• Revolutions: $N_A = 75$ and $N_B = 45$

Substituting these values into the derived equation:

$$ \frac{a_A}{a_B} = \left(\frac{75}{45}\right)^2 \left(\frac{1}{2}\right) $$

Simplifying the fraction of revolutions:

$$ \frac{75}{45} = \frac{5 \times 15}{3 \times 15} = \frac{5}{3} $$

Now, we calculate the final ratio:

$$ \frac{a_A}{a_B} = \left(\frac{5}{3}\right)^2 \left(\frac{1}{2}\right) = \left(\frac{25}{9}\right) \left(\frac{1}{2}\right) = \frac{25}{18} $$