Effective k and m — the tweak questions
Trolley between two springs (both stay stretched): displace x → one pulls harder, other pulls less → F = 2kx, so k_eff = 2k, ω = √(2k/m). Spring with mass (not light): part of the spring also oscillates → effective m larger → f lower than 2π-formula predicts. Mass removed from a spring system (washing-machine concrete): M drops → f₀ = √(k/M)/2π rises — resonance now happens at a higher rotation speed.
What this actually means
Many 'new' systems are the standard spring with a disguised k or m. Trolley between two stretched springs: displace it by x and one spring pulls harder while the other relaxes, so the net restoring force is 2kx and k_eff = 2k.
A spring that is not light oscillates part of itself along with the load, raising the effective inertia. Measured frequency comes out LOWER than the light-spring formula predicts.
Remove mass instead (the washing machine that loses its concrete block): M falls, so f₀ = √(k/M)/2π rises, and resonance shifts to a higher rotation speed. The exam question is always 'which way does f move' — trace it through the formula.
Prove it — watch it be true
- Note T with the defaults, then double the spring constant k — the period shrinks by √2, exactly what k_eff = 2k does to a two-spring trolley
- Halve the mass m instead — frequency rises as f = √(k/m)/2π predicts, the washing-machine-concrete effect
- The tweak questions are all this one readout: push k up or m down, watch f follow √(k/m)