Precision Close-Coiled Helical Spring
Provide a linear Hookean axial restoring force (k₀ = 20.00 N m⁻¹) under vertical elongation, with measured physical wire mass m_s = 60.0 g.
Physics / Loaded Spring Vibration
FREE TO EXPLOREEXPERIMENT 36 · PHYSICS LABORATORY
Harmonic Vibration of a Loaded Helical Spring and Determination of Effective Self-Mass
Determine the elastic spring stiffness constant from the period of vertical simple harmonic oscillation under varying suspended loads, and evaluate the effective dynamic contribution of the spring's distributed self-mass. Explore a virtual apparatus, collect readings and analyse a graph.
CHANGE · OBSERVE · MEASURE
Your virtual workbench.
TIME FOR 20 OSCILLATIONS T₂₀—
MEASUREMENT STATUS
Ready for a virtual observation.
Adjust the apparatus, then take a reading.
Timing is generated from the model; a short on-screen trial represents a longer physical interval. Animation is illustrative.
Simulation assumptions and limits
The teaching model assumes pure longitudinal Hookean elasticity and a linear velocity distribution along the spring wire corresponding to Rayleigh's m_eff = (1/3) m_s approximation. In a physical laboratory, small pitch-angle helix corrections, air drag damping, and slight support stand compliance introduce minute second-order deviations. The loaded spring harmonic vibration experiment provides an elegant demonstration of distributed mass mechanics in continuous systems. By combining ordinary least-squares regression with energy principles, students verify both Hooke's elastic stiffness constant k and Rayleigh's one-third kinetic energy theorem m_eff = (1/3) m_s with sub-percent precision.
Know your apparatus.
Rigid Support Stand & Suspension Clamp
Secure the top loop of the helical spring rigidly without slippage or mechanical compliance, preventing parasitic energy dissipation to the bench.
Calibrated Slotted Weight Hanger & Masses
Provide precision loading increments (100.0 g to 300.0 g, calibrated to ±0.05 g) to vary the oscillating system inertial mass.
Vertical Anti-Parallax Millimeter Scale
Verify vertical oscillation amplitude (≤ 2.0 cm) and maintain small-strain linear elastic behavior throughout all observation runs.
Precision Digital Electronic Chronometer
Measure the elapsed transit duration for 20 complete vertical cycles with 0.01 s resolution, minimizing fractional period timing error.
Fiducial Equilibrium Level Pointer
Provide a sharp horizontal reference line at static equilibrium to trigger chronometer timing precisely at the point of maximum vertical velocity.
FOLLOW THE METHOD
A reason behind every step.
Determine the elastic spring stiffness constant from the period of vertical simple harmonic oscillation under varying suspended loads, and evaluate the effective dynamic contribution of the spring's distributed self-mass. Suggested independent settings: 100.0, 150.0, 200.0, 250.0, 300.0 g.
Spring Mass Measurement and Support Mounting
Weigh the unextended helical spring on a precision laboratory balance to determine its physical wire mass m_s = 60.0 g. Clamp the upper suspension hook firmly into the rigid support stand boss-head, ensuring that the spring hangs freely without rubbing against the vertical mounting rod.
Why this step matters
Determining the physical mass of the spring wire is essential for comparison with the dynamic effective mass m_eff extracted from the regression y-intercept.
Suspended Load Attachment and Static Equilibrium Tare
Attach the weight hanger and load weights to establish the initial trial mass of m = 100.0 g. Allow the loaded spring to gently extend under gravity until it comes completely to rest at its static equilibrium position y_eq = m g / k.
Why this step matters
The system must oscillate symmetrically about its true gravitational equilibrium level, where the gravitational force is exactly balanced by initial elastic tension.
Fiducial Pointer Alignment and Amplitude Setting
Align the horizontal pointer on the weight hanger with the index mark on the vertical scale. Displace the load vertically downward by a small, controlled amplitude of approximately 1.5 cm to 2.0 cm strictly along the vertical axis, ensuring zero lateral wobble or conical swinging.
Why this step matters
Small vertical displacements guarantee that the spring remains strictly within its linear Hookean regime and prevents the excitation of parasitic pendulum or torsional modes.
Chronometer Triggering and 20-Cycle Timing
Release the load cleanly from rest without imparting any initial velocity. Allow two initial cycles to pass to ensure steady-state harmonic motion, then trigger the digital chronometer as the pointer passes upward through the static equilibrium fiducial mark. Count exactly 20 complete cycles and stop the timer, recording t₂₀ to 0.01 s.
Why this step matters
Timing 20 complete cycles reduces the fractional uncertainty of human reaction time by a factor of 20, and triggering at equilibrium maximizes timing sharpness where velocity is greatest.
Systematic Load Variation Across the Mass Range
Add slotted masses in increments of 50.0 g to test settings of 150.0 g, 200.0 g, 250.0 g, and 300.0 g. At each mass station, verify vertical alignment, maintain a constant initial displacement amplitude, and record the elapsed time for 20 complete oscillations.
Why this step matters
Collecting observations across a broad range of load masses provides the leverage required for high-precision slope and intercept determination in ordinary least-squares regression.
Period Calculation, Squaring, and Regression Analysis
Compute the single-cycle period T = t₂₀ / 20.0 and calculate T² for each load station. Plot T² (in s²) on the vertical axis against suspended load mass m (in kg) on the horizontal axis. Fit a linear regression line to determine the slope S and y-intercept b, then compute the spring stiffness k = 4π² / S and the effective self-mass m_eff = b / S.
Why this step matters
Plotting T² against m linearizes the harmonic oscillator governing equation, allowing simultaneous extraction of the elastic stiffness constant and the spring's effective self-mass.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into a result.
Points are your observations. The fit uses SI x and y values before display scaling.
LINEAR FIT · y = mx + b
T² = (4π² / k) · m + (4π² / k) · m_eff = m_fit · m + b_fit ⇔ k = 4π² / m_fit
| # | Suspended Load Mass m (g) | Time for 20 Oscillations t₂₀ (s) | Load mass m (kg) | Period squared T² (s²) | Remove |
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