I recently had the following homework problem (Knight 3e, exercise 16.72, page 467):
The cylinder in Figure CP16.72 has a moveable piston attached to a spring. The cylinder's cross-section area is $10 \, \mathrm{cm^2}$, it contains $0.0040 \, \mathrm{mol}$ of gas, and the spring constant is $1500 \, \mathrm{N/m}$. At $20 \, \mathrm{°C}$ the spring is neither compressed nor stretched. How far is the spring compressed if the gas temperature is raised to $100 \, \mathrm{°C}$?
I came up with the following solution:
There are three forces acting on the piston at any given time: $pA$ to the right, $p_{\mathrm{atmos}} A$ to the left, and $kx$ to the right. When the system is in equilibrium, $pA = p_{\mathrm{atmos}} A + kx$. At $T_1 = 293 \, \mathrm{K}$, the string is neither stretched nor compressed, so $x = 0$; further, given that the system is at equilibrium, we have $p_1 A = p_{\mathrm{atmos}} A$, so $p_1 = p_{\mathrm{atmos}} = 101.3 \, \mathrm{kPa}$. Therefore, the above invariant can be rearranged to \begin{align} p_2 &= \frac{1}{A} (p_1 A + kx). \tag{1} \end{align} Note also that \begin{align} V_2 = V_1 + Ax. \tag{2} \end{align} Then, from the ideal gas law, $p_2 V_2 = nRT_2$; substituting $(1)$ and $(2)$ gives \begin{align*} \frac{1}{A} (p_1 A + kx) (V_1 + Ax) &= nR T_2 \\ (p_1 A + kx) (V_1 + Ax) &= nR T_2 A \\ (Ak)x^2 + (V_1 k + p_1 A^2) x + A(p_1 V_1 - nR T_2) &= 0. \tag{3} \end{align*} This is a rather ugly but quite valid quadratic equation for $x$. Numerical evaluation yields a solution of $x = 1.0198 \, \mathrm{cm}$. Interestingly, there is also an equilibrium point at $x = -17.3927 \, \mathrm{cm}$, but this is longer than the depth of the cylinder and thus impossible.
I checked with my professor, and the solution manual uses a similar approach and ends up with the same equation in $(3)$.
I'm wondering what that extra, negative solution for $x$ represents. It doesn't make sense to me that there should be a negative solution: if $x$ is negative, then the spring is expanded and the gas is compressed, so there's nothing to counteract the piston's movement to the right and keep the system at equilibrium (except for the insufficiently strong atmospheric pressure).
My instructor suggested that deriving a potential energy formula $U(x)$ for the whole system might show that the desired solution is impossible to get to (i.e., would require infinite force or something because it's past a place where the required energy diverged), but I don't know how to go about doing that.
What's the significance or meaning of this solution? Is there any case where there could be a negative solution that's not longer than the length of the cylinder—and thus is a valid solution? If so, what would such a system look like?
