Spring–Mass Period versus Mass (T² vs m)

A fixed horizontal spring with unknown stiffness k; vary the attached mass m, record the small-amplitude period T with timing noise, and recover k from a linear fit of T² versus m.

School· 24 min·Related simulator: Classical MechanicsSpring-Mass System

Goal

Determine the spring constant k from the relation T² = (4π²/k)·m for undamped harmonic motion by measuring T at several masses and fitting T² as a function of m.

Equipment

Horizontal coil spring (unknown k)
Sliding mass m
Stopwatch / period readout (simulated)

Theory

For an ideal spring–mass oscillator without damping, ω₀ = √(k/m) and the period is T = 2π/ω₀ = 2π√(m/k). Squaring gives T² = (4π²/k)·m, a straight line through the origin in the (m, T²) plane with slope 4π²/k. (The companion lab “Determining g with a Pendulum” explores the analogous T²(L) law for a simple pendulum.)

Procedure

The bench hides a fixed spring constant k; you only change the slider mass m (horizontal motion, small-amplitude regime).
Pick a mass and press “Record measurement” to log (m, T, T²) with small stopwatch noise on T.
Repeat for at least 6 different masses spread between about 0.8 kg and 9 kg.
Inspect the linear fit of T² versus m: the slope equals 4π²/k, so k = 4π²/slope.
Compare your k with the reference value and write the conclusion.

Experiment

Ideal small-amplitude horizontal oscillations: T = 2π√(m/k). The bench hides k; timing noise is simulated on each recorded T.

For undamped motion, T² = (4π²/k)·m — a line through the origin with slope 4π²/k. Compare with the pendulum lab (T² vs L → g).

Oscillating mass mm = 2.50 kg

Ideal period: T = 2π√(m/k) ≈ 1.378 s

Take at least six masses between about 0.8 kg and 9 kg. Fit T² versus m; then k = 4π² / slope.

Measurements

№	Mass m kg	Period T s	T² s²
No measurements yet — take your first reading.

Data processing

Add at least 2 measurements to compute the fit.

Uncertainty

Instrument uncertainty (simple propagation)

Estimate Δm (mass setting / balance)

±0.012 kg

Estimate ΔT (timing)

±0.0020 s

δ(T²) ≈ 2·|T|·ΔT per row; mean Δm/|m| uses your Δm slider. Same spirit as the pendulum T²(L) lab.

Take measurements to estimate propagated uncertainty.

The linear fit uses all (m, T²) points; these sliders only estimate instrument-style uncertainty.

Lab report

Opens the system print dialog — choose “Save as PDF” or your printer. Header and footer are hidden when printing.

One click opens the print dialog — choose “Save as PDF”.

Spring–Mass Period versus Mass (T² vs m)

Generated: 13 May 2026, 01:04

Goal

Determine the spring constant k from the relation T² = (4π²/k)·m for undamped harmonic motion by measuring T at several masses and fitting T² as a function of m.

Measurement table

#	Mass m (kg)	Period T (s)	T² (s²)
No measurements yet — take your first reading.

Fit and derived value

Add at least 2 measurements to compute the fit.

Conclusion

The fitted k agrees with the reference stiffness within tolerance. Main uncertainties: timing noise, assuming undamped harmonic motion, and using the small-amplitude ideal period formula.

PhysSandbox virtual lab — values come from your session; add your own discussion of error sources.

Conclusion

The fitted k agrees with the reference stiffness within tolerance. Main uncertainties: timing noise, assuming undamped harmonic motion, and using the small-amplitude ideal period formula.