SHM needs a linear restoring force

01
The tempting model
Calling any back-and-forth motion simple harmonic.
02
Why it feels right
Lots of oscillations look sinusoidal at a glance.
03
Where it breaks · how to catch it
You applied ω = √(k/m) to a system whose restoring force isn't ∝ −x.
04
The correct model
SHM requires F = −kx (restoring force proportional to displacement). A pendulum is SHM only in the small-angle limit where sin θ ≈ θ.
05
Repair · see it, then test it
worked example Is a ball bouncing on the floor simple harmonic?
Tempting move
Yes — it repeats.
Correct move
No — the force isn't −kx; the motion isn't sinusoidal.
Test the repair SHM-01 · transfer

A particle feels F = −bx³. Is its full motion simple harmonic?
Yes, because the force points toward equilibrium Yes, because the force is odd in x No, because the restoring force is not proportional to x No, because acceleration must be zero

Make a fresh prediction before referring to the worked example.