The magnitude of centripetal acceleration, $a_c$, for an object in uniform circular motion can be expressed in terms of its linear velocity $v$, angular velocity $\omega$, and radius $r$ using the following key relations:

$$a_c = \frac{v^2}{r} \quad (1)$$ $$a_c = r\omega^2 \quad (2)$$ $$a_c = v\omega \quad (3)$$

We will use these formulas to find the ratio of the centripetal accelerations, $a_1 : a_2$, for the two balls under the given conditions.

[Q1] Same radii, linear velocity ratio 1:2 Given: $r_1 = r_2$ and $v_1 : v_2 = 1 : 2$, which implies $v_2 = 2v_1$. Using equation (1), the ratio of the accelerations is:

$$\frac{a_1}{a_2} = \frac{v_1^2/r_1}{v_2^2/r_2}$$

Substituting the given conditions:

$$\frac{a_1}{a_2} = \frac{v_1^2/r_1}{(2v_1)^2/r_1} = \frac{v_1^2}{4v_1^2} = \frac{1}{4}$$

Thus, the ratio $a_1 : a_2$ is $1:4$.

[Q2] Same linear velocities, radii ratio 1:2 Given: $v_1 = v_2$ and $r_1 : r_2 = 1 : 2$, which implies $r_2 = 2r_1$. Using equation (1):

$$\frac{a_1}{a_2} = \frac{v_1^2/r_1}{v_2^2/r_2}$$

Substituting the given conditions:

$$\frac{a_1}{a_2} = \frac{v_1^2/r_1}{v_1^2/(2r_1)} = \frac{1/r_1}{1/(2r_1)} = 2$$

Thus, the ratio $a_1 : a_2$ is $2:1$.

[Q3] Same angular velocities, radii ratio 1:2 Given: $\omega_1 = \omega_2$ and $r_1 : r_2 = 1 : 2$, which implies $r_2 = 2r_1$. Using equation (2), which is more direct for this case:

$$\frac{a_1}{a_2} = \frac{r_1\omega_1^2}{r_2\omega_2^2}$$

Substituting the given conditions:

$$\frac{a_1}{a_2} = \frac{r_1\omega_1^2}{(2r_1)\omega_1^2} = \frac{1}{2}$$

Thus, the ratio $a_1 : a_2$ is $1:2$.

[Q4] Linear velocity ratio 1:2, angular velocity ratio 2:3 Given: $v_1 : v_2 = 1 : 2 \implies v_2 = 2v_1$, and $\omega_1 : \omega_2 = 2 : 3 \implies \omega_2 = \frac{3}{2}\omega_1$. Using equation (3), which directly relates the given quantities:

$$\frac{a_1}{a_2} = \frac{v_1\omega_1}{v_2\omega_2}$$

Substituting the given conditions:

$$\frac{a_1}{a_2} = \frac{v_1\omega_1}{(2v_1)(\frac{3}{2}\omega_1)} = \frac{v_1\omega_1}{3v_1\omega_1} = \frac{1}{3}$$

Thus, the ratio $a_1 : a_2$ is $1:3$.