C

To determine when the coin will rotate with the gramophone, we need to analyze the forces acting on the coin.
Step 1: The maximum static friction force that can act on the coin is given by:
$$ F_{f} = \\mu m g $$
where $\mu$ is the coefficient of static friction, $m$ is the mass of the coin, and $g$ is the acceleration due to gravity.
Step 2: The centripetal force required to keep the coin rotating in a circle of radius $r$ at angular velocity $\omega$ is given by:
$$ F_{c} = m\frac{\omega^2 r}{g} $$
Step 3: For the coin to rotate with the gramophone, the maximum static friction must be equal to or greater than the centripetal force required. Thus, we set:
$$ \mu m g \geq m \frac{\omega^2 r}{g} $$
Step 4: Dividing through by $m$ (assuming $m \neq 0$) and rearranging gives:
$$ r \leq \frac{\mu g}{\omega^2} $$
Thus, the condition for the coin to rotate with the gramophone is that the distance $r$ must be less than or equal to $\frac{\mu g}{\omega^2}$. Therefore, the correct option is C.