Linear acceleration on a spinning satellite with an unbalanced force If there is a satellite in orbit in space, with an off centre booster providing an unbalanced force, it will experience rotational acceleration. However, I was wondering if it also undergoes linear acceleration while spinning, or only for a brief moment at the start...
 A: 
Here I have drawn a simple diagram of what might answer your question. $\overrightarrow{a}$ is the acceleration provided by the off-centred booster. $\overrightarrow{r}$ is a vector from centre-of-mass of the rocket to the point at which booster is applying force.
We divide, for our convenience, $\overrightarrow{a}$ into it's components along $X$ and $Y$ axes, where $X$-axis is perpendicular to $\overrightarrow{r}$.
Let the $x$-component of acceleration be $\overrightarrow{a_x}$ and $y$-component be $\overrightarrow{a_y}$.
Now you can easily spot which component provides linear acceleration ($\overrightarrow{a_y}$) and which one provides rotational acceleration ($\overrightarrow{a_x}$).
So, answering your original question - It undergoes both linear and rotational acceleration (throughout the burnout of the booster). The satellite would be linearly accelerated the whole time, although not along the direction of it's nose.
Edit: Please note that $\overrightarrow{r}$ and $\overrightarrow{a_y}$ are overlapping each other since $\overrightarrow{r}$ $\perp$ $X$-axis and $Y$-axis $\perp$ $X$-axis. They have been drawn a bit off in the diagram by mistake.  
