# Unsupported rotating ring speed limit

 The maximum speed any physical thing made out of atoms can be spinned is about 3000 meters per second. How natural accelerations grow with shrinking size. Offtopic: To keep waste heat from friction at practical levels it is sensible to slow down at the nanoscale that is as one goes from right to left in the diagram one moves down the lines deviating from the natural scaling law. (TODO: make a simplified graph with just the red line for this page)

For every material made into a ring there is a maximum tangential speed it can be rotated before the forces rip it apart. This speed depends on the materials ultimate tensile strengh (UTS). As it turns out this speed is independent of the size of the ring. Since in the limits of currently known physics there is an upper limit in material strengths that are reachable there is an upper limit for rotating speed.
Using one of the materials with highest known UTS (carbon nanotubes) one finds that:
This limit is at about 3000 m/s

## A practical problem where the limit might matter

In certain situations the unsupported rotating ring speed limit is a factor that plays a role in the absolute limit of possibly transmittable power density in mechanical energy transmission cables.

(TODO: add the math)

## A practical problem where the limit does not apply

If infinitesimal bearings are arranged in a straight track there are no forces so the limit does not apply. So in the not so near future Interplanetary acceleration tracks may be buildable that catch spacecraft not electromagnetically but physically and that at orbital speeds.

## Exceeding the limit

To exceed this rotation speed the ring would need to be supported by an enclosing non-rotating ring. Since the relative motion is about ten times thermal motion at room temperature macroscopically thick (a few millimeters) layers of infinitesimal bearing are needed that can deal with high levels of stress and deal with a bit of (reversible) strain. Even then the friction heat might be so high that these super-limit speeds can be sustained only briefly.

It is still highly unclear whether a ultra low friction levitation could be archived that would provide enough supporting force.

## Math

### Prerequisite textbook physics math

• (1) $F = m a$ Newton
• (2) $m = \rho V = r^3 C_1$ mass from density and characteristic size
• (3) $a = \omega^2 r$ magnitude of centrifugal and virtual rotating frame acceleration
• (4) $\omega = v / r$ when going through a half sized cycle of movement with the same speed it doubles the frequency
• (5) $A = r^2 C_2$ the area that force is acting on
• (6) $E = F/A$ stress on the rotating ring
• (7) $E \lt E_{max}$ condition for a specific material to not rupture under stress

$C_1$ and $C_2$ are just geometry specific constants.

### Putting things together

• (8) $F = m a = m \omega^2 r = (r^3 C_1)*((v/r)^2 r)*(r) = v^2 r^2 C_1 = F$ Used: (1) (2) (3) (4)

Centrifugal forces scale quadratically with size (given constant speeds)
That is: Half the size gives quarter the force

• (9) $E = F/A = v^2 r^2 C_1 / (r^2 C_2) = v^2 C_{1,2} \lt E_{max}$ Used: (6) (7) (8)

Result: Centrifugal stresses are scale invariant (assuming rotation speeds are kept constant)
That is: Half the size gives the same stress.

And that is why for one ultimate maximal tensile material strength there is just one unsupported rotating ring speed limit.