Consider a semi pulling a tractor trailer–if the truck turns, will the trailer straighten out completely? My friends and I are having a very heated debate about this question: Under perfect conditions (i.e. only considering friction from the road and no other forces), if a semi is pulling a tractor trailer and turns, would the trailer straighten out completely after the turn or would it asymptotically approach straightness (and why)? This question is probably way too vague, so if I need to include any additional information let me know.
 A: You can model this system as a mass connected to a spring and damper in parallel:

The mass represents the inertia of the trailer as it swings in an angular fashion behind the cab. The spring represents the driving force for the trailer to move straight, since the orientation of the wheels makes this direction strongly preferred. The damper represents the dissipative action of friction.
(Please note that I'm not claiming that the spring is a linear spring or the damper a linear damper over any large range; I'm only presenting a system that captures the key elements of your chosen level of idealization: the existence of friction and the consideration of the mass of the trailer, but not the existence of wind or pebbles on the road that would add stochastic disturbances, for example. In real life, small disturbances would ensure that the trailer eventually crosses the center line.)
This type of system has been thoroughly studied (e.g., here) and is known to exhibit underdamped, overdamped, and critically damped solutions depending on the characteristic values of the lumped components. In your example, an underdamped trailer would cross the center line and subsequently oscillate around it; an overdamped or critically damped trailer would not cross but would rather asymptotically approach the center line. (The critically damped system is distinguished by exhibiting the fastest possible approach to the centerline without crossing it.)
Determining the type of behavior that your thought experiment would exhibit therefore requires more precise (e.g., quantitative) definition of the characteristics of the system. The question cannot be answered definitively as stated.
A: As mentioned in the other answer; it should behave as some sort of spring-damper system.
I disagree with their analysis that we do not have enough information though.
We should consider the 3 types of damped motion.  Under-damped, critically damped, and over-damped.
For overdamped and critically damped systems, the control feels more "tight" and the trailer would approach it's straight position without ever turning too far past it.  This approach can be quickened; but it will still always take an infinite amount of time to reach it's equilibrium straight position,  according to the governing differential equations.
The other case is when the system is under-damped, in which the control would feel more "loose".  After making the turn, the trailer would turn too much and have to turn the opposite way to correct itself.  This will continue happening back and forth.  The frequency and amplitude of that oscilation around the middle will vary; but just as in over and critically damped systems, the oscillations will never truly settle on the middle.
Essentially, in all scenarios the trailer will only ever asymptotically approach being straight.  Depending on the specific setup, there are multiple ways this asymptotic approach can appear.
If it is critically or over-damped, it will asymptotically approach from one side only without ever crossing.
If it is under-damped it will oscillate side to side to straighten itself, with the oscillation amplitudes asymptotically approaching a straight path.
