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.
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.
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
Uncertainty
Instrument uncertainty (simple propagation)
±0.012 kg
±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: 23 Apr 2026, 05:27
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.