1. Identify the force providing circular motion

A coin placed on a rotating disc moves in a circle of radius $r$ with the disc, provided it does not slip.

For circular motion, the required centripetal force is $F_c = m r \omega^2$ where $m$ is the mass of the coin and $\omega$ is the angular velocity.

2. Source of centripetal force

The only horizontal force acting on the coin is friction.

So, static friction must provide the required centripetal force: $f = m r \omega^2$

3. Maximum available static friction

The maximum static friction is $f_{\max} = \mu N$ Since the disc is horizontal, $N = mg$ Therefore, $f_{\max} = \mu mg$

4. Condition for no slipping

For the coin not to slip, $m r \omega^2 \le \mu mg$ Cancel $m$ from both sides: $r \omega^2 \le \mu g$

So the maximum angular velocity is obtained when equality holds: $r \omega_{\max}^2 = \mu g$

Hence, $\omega_{\max} = \sqrt{\frac{\mu g}{r}}$

5. Match with options

This corresponds to: $\boxed{\text{B: } \sqrt{\frac{\mu g}{r}}}$

6. Comparison with stored answer

Stored correct answer: B

Our derived answer: B

So, the derived answer agrees with the stored answer.