Derive spring topology and shift the equilibrium coordinate

Derive equivalent stiffness from constraints

Two parallel springs undergo the same extension x and provide forces k1x and k2x; their total force is (k1+k2)x. For massless springs in series, the same force F acts through each, while total extension is F/k1+F/k2. Thus k_parallel=k1+k2 and 1/k_series=1/k1+1/k2. These results depend on the stated geometry and ideal massless connectors, not merely on seeing two springs.

Translate a vertical force law

Choose downward y measured from the unstretched configuration. For attached mass m, m y double-dot=mg−keff y. Equilibrium is y_eq=mg/keff. With q=y−y_eq, the equation becomes m q double-dot=−keff q. Gravity supplies a constant offset but does not change the linear restoring slope. The frequency is sqrt(keff/m), not sqrt(g/y) using an arbitrary instantaneous extension.

Predict changes before computing

Increasing mass lengthens the period by its square-root factor. Increasing effective stiffness shortens it. Under the same mass, parallel springs are stiffer than either alone and series springs softer than either alone. Displacement must be measured from each configuration’s own equilibrium before comparing restoring forces; the unstretched reference and equilibrium reference are different.

State the limits of the equivalent model

The derivation assumes linear spring response, negligible spring/connector masses and geometry that remains fixed. A heavy spring has distributed motion and a constrained support can add other modes. The workbench stipulates enough linear extension range for every virtual configuration; it does not claim that a real spring can safely stretch to the computed equilibrium.

Worked original case

For k1=k2=8 N/m and m=2 kg, the single, parallel and series stiffnesses are 8,16,4 N/m. Their equilibrium extensions are 2.5,1.25,5 m in the stipulated ideal model. Angular frequencies are 2,sqrt(8),sqrt(2) rad/s; the large extensions are mathematical parameters, not apparatus instructions.

Investigation

Derive all three stiffnesses for two different k2 values. Compute each equilibrium first, then compare periods at the same mass. Double mass in a written changed case and predict the period factor before calculating. Draw which quantity is shared in a series connection and which in a parallel connection.

Public numerical transfer practice

Try a new practice case

Two ideal springs of stiffness 6 and 3 N/m are in series. What is their effective stiffness in N/m?

Numerical result in the requested units Check this practice result

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Sources and evidence boundary

Original derivations and assigned numerical cases checked 30 October 2026 against the official AP Physics C Mechanics framework and OpenStax harmonic motion and pendulums. These are supplied or generated records, not physical measurements or performed laboratory time. Do not construct rotating equipment, release loads or conduct launch experiments. Public feedback and retained self-reports do not certify independent mastery.