The relationship between centripetal acceleration ($a_c$) and radius ($r$) is not unique but depends on which physical quantity is held constant while the radius is varied. Both students' reasoning is valid, but each applies to a different physical situation.

The two common formulas for the magnitude of centripetal acceleration are derived from each other using the relationship between linear speed ($v$) and angular speed ($\omega$), which is $v = \omega r$.

Starting with the expression in terms of linear speed:

$$a_c = \frac{v^2}{r}$$

We can substitute $v = \omega r$ to obtain the expression in terms of angular speed:

$$a_c = \frac{(\omega r)^2}{r} = \frac{\omega^2 r^2}{r} = \omega^2 r$$

Both formulas are physically correct and equivalent. The confusion arises from the interpretation of proportionality, which requires holding other variables constant.

Scenario 1: Constant Linear Speed ($v$ is constant) This scenario corresponds to Student 1's reasoning. Using the formula $a_c = v^2/r$, if $v$ is held constant, then $a_c$ is inversely proportional to $r$.

$$a_c \propto \frac{1}{r} \quad (\text{for constant } v)$$

An example is a car maintaining a constant speed while navigating circular turns of different radii. A larger radius turn requires less centripetal acceleration.

Scenario 2: Constant Angular Speed ($\omega$ is constant) This scenario corresponds to Student 2's reasoning. Using the formula $a_c = \omega^2 r$, if $\omega$ is held constant, then $a_c$ is directly proportional to $r$.

$$a_c \propto r \quad (\text{for constant } \omega)$$

An example is points on a spinning turntable. All points have the same angular speed, but points farther from the center (larger $r$) experience a greater centripetal acceleration.

Therefore, the correct answer depends entirely on the physical constraints of the system in question.