Consider a body inside an elevator. If the elevator is going at a constant speed and starts to accelerate or deccelerate, a body inside it perceives the resulting inertia as a change in $g$. Now, let us attach this body to the elevator with a spring. What would be its equation of motion?
Normally, a motion of a weight on a spring is described by a well known equation:
$$\frac{d^2x}{dt^2} + \frac{k}{m}x=0$$
And its general solution is:
$$x=Acos(\omega_0t)+Bsin(\omega_0t),\qquad\omega_0=\sqrt{k \over m}$$
This equation is valid in a gravitational field although it does not take $g$ into account. The question is: will it be valid, it $g$ is not constant but varies in time, like in an elevator or another inertial frame?
On one hand, the equation does not make any assumption about the acceleration function, it is simply a second derivative of $x$, so it seems that the solution will be valid.
On the other hand, in terms of forces, varying acceleration at constant mass is equivalent to varying mass at constant acceleration. So, if mass varies $\omega_0$ is no longer constant in time and the equation has probably a totally different solution. What would it be?