# Relationship between how fast a metal cylinder can spin on its axis without breaking or deforming?

For simplicity sake consider a solid metallic cylinder which is $$2r$$ in height with a radius of $$r$$.

For a given metal and $$r$$, how to calculate how fast can this spin in terms of rpm for a very long time without breaking or deforming?

Also which is the best metal to achieve huge rpm without breaking or deforming?

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• Do some research on flywheels storing energy. Aug 10, 2023 at 10:55
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• Such an easy topic to get good results. Aug 10, 2023 at 10:57

For a thin spinning disk ignoring the frictions, such as air friction, there are two stresses, radial and hoop stress. Both must be below the accepted stress limit of the material. We assume no vibration and warping or fluttering.

stress formula is

$$σ_r = \frac{\rho \omega^2 r}{t}$$

• $$\sigma_r$$ is the radial stress (in Pascals or N/m²),
• $$\rho$$ is the material density, kg/m^3
• $$\omega$$ angular velocity, radians/s
• t thickness, meters

# hoop

hoop stress is tangential to the circumference of the disk and acts along its thickness. It is caused by the resistance of the material to being stretched by the centrifugal force

$$σ_h = \frac{\rho^2 \omega^2 r^2}{2t}$$

• The question specifies a thick disk, not a thin disk. Aug 12, 2023 at 2:20