\(g^2_{net}=g^2_Tsin^2(\theta)+(g_Tcos(\theta)-g)^2\)
when \(g_T\) is large compared to \(g\)
\(g^2_{net}=g_T^2sin^2(\theta)+g^2_Tcos^2(\theta)\)
then
\(g^2_{net}=g_T^2\)
which is a sphere in 3D (polar co-ordinates). It is still best to use the solid generated by rotating the curve, given in that post, about the vertical axis.
From the post "Contained But Not Stressed", we have the thickness of such a shell given by,
\(\cfrac{L_e}{L_i}=\sqrt{\cfrac{\Delta L|_{T_{max}}}{\Delta L|_{T_{o}}}}\)
where the material is chosen such that \(T_{max}<T_{melting}\); the temperature when matter/antimatter annihilation occurs is just less than the melting temperature of the material.
From the post "kaBoom',
\(F_T=\cfrac{4\sqrt{3}}{3}\pi\rho L^2 D.\cfrac{\partial (d_s)}{\partial T}.\cfrac{1}{\alpha}\)
We set \(L=L_i\) since \(T_{max}>T_{o}\) because of the temperature gradient between the inner and outer wall of the containment shell. The Young's Modulus of the material must be such that, the yield point of the material, \(F_y\) is,
\(F_T=\cfrac{4\sqrt{3}}{3}\pi\rho L^2_i D.\cfrac{\partial (d_s)}{\partial T}|_{T_{max}}.\cfrac{1}{\alpha}<F_y\)
for effective containment. From the post "Young And On Heat"
\(F_L=E_s\cfrac{A_o}{L_o}\Delta L\)
In general for a sphere of radius \(L_i\), \(L_o=L_i\) and \(A_o=4\pi L^2_i\),
\(F_L=4\pi E_sL_i\Delta L=F_T=\sqrt{3}\rho D.\cfrac{\partial (d_s)}{\partial T}.\cfrac{1}{L_i\alpha}\)
where the lattice force, \(F_L\) as a result of \(\Delta L\), counteracts the force due to thermal gravity, \(F_T\) when the sphere is heated. Since,
\(E_s=\sqrt{3}{E}\)
\(D.\cfrac{\partial (d_s)}{\partial T}=4\pi E\cfrac{\alpha}{\rho}L^2_i\Delta L\)
\(D.\cfrac{\partial (d_s)}{\partial T}|_{T_{max}}=4\pi E\cfrac{\alpha}{\rho}L^2_i\Delta L|_{T_{max}}\)
substitute into the expression for \(F_y\),
\(\cfrac{16\sqrt{3}}{3}\pi^2E.L^4_i\Delta L|_{T_{max}}<F_y\)
where \(\Delta L|_{T_{max}}\) is the increase in radius of a sphere originally of radius \(L_i\), when temperature is at \(T_{max}\) uniformly, and \(E\) is the Young's Modulus.
And so we have,
The outermost sphere is an inductive coil to heat the interior of the containment shell. \(f_{res}\) can be increased by spinning the induction sphere but not necessarily. \(f_{res}\) depends on the element contained inside and can be found somewhere in this blog. No, we are not melting the element inside; if stupid, use the other force. The point is to have the heat in the shell, from matter/antimatter annihilation, melts the containment instantaneously and release all contained energy in an instance. The containment shell can be sealed by friction welding two hemispheres with the heavy element of large \(r_{n}\) inside.
Have a big Kaboom on me!
