Double-spring mass system

Question

We just had a lesson about elementary mass-spring systems (SHO), and I thought about a horizontal situation with two springs with the test mass oscillating in between. If we are to manually stretch the mass a distance $\Delta x<\Delta_0$ , we obtain opposing restoring forces, which yields the familiar SHO equation. But do thesee conditions change when we stretch it beyond $\Delta_0$(a little beyond, it's still elastic with Hooke's law), in some intervals the restoring forces point in the same direction and in others don't. Does the system retain its "SHO-ness"?

Answer 1

The system does remain a simple harmonic oscillator. Consider an arbitrary upward displacement from equilibrium of $d$, where $d > 0$. The top spring will be compressed, so it will push downward on the block with force $F_{top} = - k d$. The bottom spring will be extended, so it will pull downward on the block with force $F_{bot} = - k d$. Overall, since $F_{top}$ and $F_{bot}$ are in the same direction, $F_{net} = -2kd$. A similar analysis yield an identical result for a downward displacement ($F_{net} = -2kd$, with $d < 0$).

Therefore, force is proportional to displacement, and the motion is simple harmonic.