From the post "Hollow Earth",
\(F=-\cfrac { GM }{ 4Rr^{ 2 } } \int _{ s=R-r }^{ s=R+r }{ \left( 1+\cfrac { r^{ 2 }-R^{ 2 } }{ s^{ 2 } } \right) } ds.m\)
\( g(s)=\cfrac { F }{ m } =-\cfrac { GM }{ 4Rr^{ 2 } } \int { \left( 1+\cfrac { r^{ 2 }-R^{ 2 } }{ s^{ 2 } } \right) } ds\)
On the outer surface of hollow earth,
\( s=r-R\), \(s\gt0\)
\( \cfrac { d\, g(s) }{ d\, s } =-\cfrac { GM }{ 4Rr^{ 2 } } \left( 1+\cfrac { r^{ 2 }-R^{ 2 } }{ s^{ 2 } } \right) +\cfrac { GM }{ 2Rr^{ 3 } } \int { \left( 1+\cfrac { r^{ 2 }-R^{ 2 } }{ s^{ 2 } } \right) } ds\)
\( =-\cfrac { GM }{ 4Rr^{ 2 } } \left\{ 1+\cfrac { r^{ 2 }-R^{ 2 } }{ s^{ 2 } } -\cfrac { 2 }{ r } \int { \left( 1+\cfrac { r^{ 2 }-R^{ 2 } }{ s^{ 2 } } \right) } ds \right\} \)
\(=-\cfrac { GM }{ 4R(s+R)^{ 2 } } \left\{ 2+\cfrac { 2R }{ s } -\cfrac { 2 }{ s+R } \int { \left( 2+\cfrac { 2R }{ s } \right) } ds \right\} \)
\(=-\cfrac { GM }{ 2R(s+R)^{ 2 } } \left\{ 1+\cfrac { R }{ s } -\cfrac { 2 }{ s+R } \int { \left( 1+\cfrac { R }{ s } \right) } ds \right\} \)
The gradient of gravity \(g^{'}(s)\) has a discontinuity at \(s=0\) due to the \(\cfrac{R}{s}\) term. This high value in gradient results in a sharp increase in gravity \(g\rightarrow 2g\) at \(s=0^{+}\), on the outer surface of hollow earth.
In reality, gravity doubles only after some distance from sea level. The difference between \(s=0\) and \(s=0^{+}\) is significant. And the next rocket launch should be less shaky.
