A block of mass m is placed on a horizontal ruler. The coefficient...

When the ruler rotates in the horizontal plane, the block moves in a circular path of radius $L$.

The necessary centripetal force for this circular motion is provided by the static friction between the block and the ruler.

The centripetal force required is $F_c = m \omega^2 L$.

The maximum static frictional force available is f_\max = \mu N = \mu m g.

For the block not to slip, the required centripetal force must be less than or equal to the maximum static friction:

$m \omega^2 L \le \mu m g$

$\omega^2 \le \dfrac{\mu g}{L}$

$\omega \le \sqrt{\dfrac{\mu g}{L}}$

Thus, the maximum angular speed is $\sqrt{\dfrac{\mu g}{L}}$.