With regards to this thread, Thrust center in space
My question is, if the thrust to weight ratio was increased so that it was much higher than the weighted mass of the sphere (ship), would the sphere then not start to follow a straight trajectory along the axis of the thruster.
As can be inferred from the Wikipedia article on center of percussion, the governing equations of a rod of mass $m$ and moment of inertia $I$ with a rocket of force $F$ attached perpendicular to the rod at a distance $b$ from the center of mass is $$m\ddot{\mathbf{r}}=R_\theta\mathbf{F}\\Fb=\ddot{\theta}$$ where $$\mathbf{F}=(0,F)$$ is the force vector applied by the rocket when $\theta=0$, where $\theta$ is the angle the rod makes with the $x$-axis, and $R_\theta$ is the 2D counterclockwise rotation matrix of angle $\theta$.
With initial conditions $\mathbf{r}(0)=\dot{\mathbf{r}}(0)=(0,0)$ and $\theta(0)=\dot{\theta}(0)=0$ this gives $$\theta(t)=\frac{b F t^2}{2 I}$$ $$\mathbf{r}(t)=\left(\frac{-\sqrt{\pi } t \sqrt{b F I} S\left(\sqrt{\frac{b F}{I\pi}} t\right)+I \left(-\cos \left(\frac{b F t^2}{2 I}\right)\right)+I}{b m},\frac{\sqrt{\pi } t \sqrt{b F I} C\left(\sqrt{\frac{b F}{\pi I}} t\right)-I \sin \left(\frac{b F t^2}{2 I}\right)}{b m}\right)$$
where $C$ and $S$ are the Fresnel C and Fresnel S functions.
Notice that the expression is invariant under the transformation $F\rightarrow\lambda F$, $t\rightarrow \lambda^{-1/2}t$. This means that the path is unchanged when you alter the thrust to weight ratio.
In short, changing the thrust to weight ratio actually does not cause the trajectory of the off-center rocket-rod to change, despite the initial physical intuition that this would cause the system's path to "straighten out".
For reference, here is a plot of the center of mass trajectory for some particular choice of constants:
ParametricPlot[{(
i - i Cos[(b F t^2)/(2 i)] -
Sqrt[b] Sqrt[F] Sqrt[i] Sqrt[\[Pi]]
t FresnelS[(Sqrt[b] Sqrt[F] t)/(Sqrt[i] Sqrt[\[Pi]])])/(b m), (
Sqrt[i] (Sqrt[b] Sqrt[F] Sqrt[\[Pi]]
t FresnelC[(Sqrt[b] Sqrt[F] t)/(Sqrt[i] Sqrt[\[Pi]])] -
Sqrt[i] Sin[(b F t^2)/(2 i)]))/(b m)} /. {i -> 1, b -> 1,
F -> 2, m -> 1}, {t, 0, 5}]
