Derive the formula for the maximum allowable rotational speed ($N_{max}$) of an external spiral retaining ring before it expands out of its groove.

The centrifugal force acting on a rotating ring tends to expand it. The critical speed is reached when the centrifugal force equals the ring's radial grip on the shaft. The formula is $N_{max} = \sqrt{\frac{4.48 \cdot 10^9 \cdot E \cdot I \cdot g}{\rho \cdot A \cdot R^4 \cdot (1+n)^2}}$ where $E$ is the modulus, $I$ is the moment of inertia, $\rho$ is the density, $A$ is the cross-sectional area, and $R$ is the mean radius. For high-RPM applications like electric vehicle motors, we use 'Self-Locking' spiral rings. These rings have a small tab that locks into a notch on the adjacent turn, mechanically preventing the ring from expanding due to centrifugal forces, thus allowing speeds $3-5$ times higher than standard rings.