110
A. K. Misra and S. Cohen
Fig. 4 A schematic diagram
of the space elevator
Fig. 5 Free-body diagram
of a ribbon element
(dm) a r =
F r = T (r + dr) − T (r ) − dF g
(7)
Substituting values for tension, gravitational force, and acceleration,
dm(−Ω
2 r ) = σ 0 A(r + dr ) − σ 0 A(r ) − dm(μ/r
2
)
(8)
where dm is the product of the bulk density, and the infinitesimal volume given by
A ds and μ is the gravitational constant of the Earth. The spatial coordinate r and
the ribbon coordinate s are related by
r = R E + s + u 0 (s) = R E + s(1 + ε 0 )
(9)
where R E is the radius of the Earth. Making the preceding substitutions and then
simplifying, Eq. (8) becomes
γ A(s) ds
μ/
R E + s(1 + ε 0 )
2 − Ω
2
R E + s(1 + ε 0 )
= σ 0 dA
(10)
A. K. Misra and S. Cohen
Fig. 4 A schematic diagram
of the space elevator
Fig. 5 Free-body diagram
of a ribbon element
(dm) a r =
F r = T (r + dr) − T (r ) − dF g
(7)
Substituting values for tension, gravitational force, and acceleration,
dm(−Ω
2 r ) = σ 0 A(r + dr ) − σ 0 A(r ) − dm(μ/r
2
)
(8)
where dm is the product of the bulk density, and the infinitesimal volume given by
A ds and μ is the gravitational constant of the Earth. The spatial coordinate r and
the ribbon coordinate s are related by
r = R E + s + u 0 (s) = R E + s(1 + ε 0 )
(9)
where R E is the radius of the Earth. Making the preceding substitutions and then
simplifying, Eq. (8) becomes
γ A(s) ds
μ/
R E + s(1 + ε 0 )
2 − Ω
2
R E + s(1 + ε 0 )
= σ 0 dA
(10)