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Physics
Classical Physics
Mechanics
Max tip speed of a spinning cable
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[QUOTE="mfb, post: 6469246, member: 405866"] It's surprising but you can do all this with pen and paper. All we need is material constants and the effective potential difference. For the space elevator this is the sum of gravitational potential and centrifugal potential, for SpinLaunch it's only the centrifugal potential ##V = \int F dr = \frac 1 2 \omega^2 r^2 = \frac 1 2 v^2## (looks familiar?). Consider an infinitesimal element of a tapered design that needs to support a tension T pulling from the outside. We allow a material- and engineering-dependent maximal tensile strength S, so we need a cross section of A=T/S which comes with a material density of ##A\rho##. That means tension increases (with decreasing radius) by ##\frac {dT}{-dr} = A \rho \omega^2 r = A \rho \frac{dV}{dr}## or ##\frac{dT}{dr} = -\frac{T \rho}{S} \frac{dV}{dr}##. Mathematicians will hate this step, but in physics we can rewrite this as ##\frac{dT}{dV} = -\frac{T \rho}{S}## and solve: ##T(V) = T_0\exp(-V\rho/S)##. Here ##\frac{\rho}{S}## is just the inverse specific strength. Let's plug in numbers: v=2000 m/s, 3700 kJ/kg specific strength of Zylon without safety factor -> [url=https://www.wolframalpha.com/input/?i=exp%281%2F2*%282000+m%2Fs%29%5E2%2F%283700+kJ%2Fkg%29%29]tapering ratio of 1.7[/url]. If we load it with 2500 kJ/kg only to have some safety factor the tapering ratio increases to 2.2. These are perfectly reasonable values and you can even go with larger safety factors (or higher speed, but you still have the atmosphere as problem). [/QUOTE]
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Max tip speed of a spinning cable
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