# Tension of a Spinning Ring

fun with physics

What is the tension in a spinning ring, for example Bank's orbitals or a Niven Ring?

Let us break the ring into *N* sections. The angle (in radians) subtended by each section, dθ, is 2π / *N*. Suppose the radius of the ring is *R* and the mass-per-unit-length of the ring is λ; then the mass of a section is λ *R* dθ and the mass of the whole ring *M* = 2π_R_λ.

Ignoring gravity, there are two forces pulling on each small section: tension pulling to the left and tension pulling to the right. Because the ring is circularly symmetric, the magnitude of the two tensions are equal. The directions of the two forces are dθ from being opposite.

If at some moment in time, the left tension, * Tl*, is pointing in the -

*direction, then the right tension will be*

**x̂**

=TrTsin(dθ)+ŷTcos(dθ)x̂

where *T* is the magnitude of either tension.

As *N* gets large: dθ gets small, cos(dθ) → 1, and sin(dθ) → dθ. So:

= -TlTx̂

= +TrT+x̂Tdθŷ

Consequently, the centripetal force on the small section is _T_dθ. From Newton's second law:

∑

F=ma

+Tl=Trma

Tdθ=ŷma

Let *a* = |* a*|, the centripetal acceleration.

Tdθ = (λRdθ)a

T= λRa

Part B: Plug in some numbers.

Let *S* be the specific strength, which is the maximum tension divided by density: *T* / λ.

S≥Ra

R≤S/a

Let *P* be the period of revolution. Since centrepetal acceleration, *a* = v² / *R*.

a= v² /R

a= (2πR/P)² / R

a= 4π²R/ _P_²

a_P_² = 4π²R

a_P_² ≤ 4π² (S/a)

_P_² ≤ 4π²

S/ _a_²

P≤ 2π √(S) /a

*Assume our desired centripetal acceleration is an Earthly 10 m/s².*

If the ring is made of solid steel, *S* = 154000 N·m/kg (specific strength). With *a* = 10 m/s², we find that the maximum radius for a ring of steel is 15 kilometers. Period of revolution = 243 seconds = 4 minutes. If you assume that half of the mass of your ring is structural material and the other half is nonstructural, then you get a diameter of 15 km. Multiply the period by 1/√2.

Average T / λ | Radius | Period | Comments |
---|---|---|---|

8 745 m²/s² | 893 m | 60 s | Stanford Torus |

40 000 m²/s² | 4 000 m | 125 s | O'Neill cylinder |

77 000 m²/s² | 7 700 m | 174 s | 50% steel |

154 000 m²/s² | 15 400 m | 246 s | Pure steel |

48 000 000 m²/s² | 4 800 000 m | 4353 s | carbon nanotubes |

24 000 000 m²/s² | 2 400 000 m | 3078 s | 50% carbon nanotubes |

Carbon nanotubes: maximum size of 4800 kilometers. Period of revolution = 4353 seconds = 1.2 hours

The period of revolution for a Culture Orbital is a convenient 24 hours. The radius of a Niven Ring is 1 AU. Neither of these is feasible with today's understanding of physics.

On the other hand, a Stanford Torus with a radius of 893 meters and an orbital period of 60 seconds would need to have an average specific strength of 8745 N·m/kg, so at least 6% of the mass would need to be steel structure. (You would probably want a 4x fudge factor. Say 25% structural steel. Then sections of the ring could be replaced if they show signs of wear.)