Loaded Spring Vibration lab report

1. Title and aim

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.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Spring Stiffness Constant of 19.98 N m⁻¹, differing by 0.081% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• 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.
• 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.
• Vertical helical spring possessing linear Hookean elasticity (reference k₀ = 20.00 N m⁻¹).
• Physical spring wire self-mass m_s = 60.0 g (effective dynamic mass m_eff = (1/3) m_s = 20.0 g).
• Timing is recorded over 20 complete vertical cycles and rounded to 0.01 s.

4. Theory and graph choice

Connect the kinematics of simple harmonic motion with the distributed mechanics of elastic helical springs. By suspending calibrated loads from a helical spring and timing 20 complete vertical oscillations, you will calculate the vibration period T, construct a linear regression of T² against load mass m, extract the spring stiffness constant k from the slope, and determine the effective self-mass m_eff from the y-intercept, validating Rayleigh's one-third kinetic energy correction.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units s² kg⁻¹.

5. Method

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0815%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 1.4466e-07; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 0.00139. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 2.9461e-04.

Propagating only the relevant fit coefficient gives a fit-only standard error of 0.01405 N m⁻¹. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Timing reaction time and visual index alignment latency at the equilibrium mark (±0.15 s over 20 cycles, ~0.008 s/cycle), small vertical amplitude control (ensuring displacement stays ≤ 2 cm to avoid exciting non-linear spring softening or lateral pendulum modes), slotted weight mass calibration tolerance (±0.05 g), helix pitch angle geometry corrections, air drag damping causing slight cycle-to-cycle amplitude decay, and clamp compliance at the rigid overhead suspension bracket contribute experimental uncertainty. The OLS regression standard error reflects random scatter across load masses, but does not capture shared scale calibration offsets or residual damping effects.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• Displace the weight hanger vertically downward strictly along the central longitudinal axis of the spring to avoid exciting lateral pendulum swinging or torsional twisting modes.
• Do not exceed the elastic limit of the spring by adding excessive load masses that would cause permanent plastic deformation of the steel coils.
• Discard the first two transient cycles after releasing the load to allow steady-state harmonic motion to establish before starting the chronometer.
• Use the fiducial equilibrium index mark to trigger the timer at the midpoint of the swing where velocity is highest, minimizing timing trigger latency error.

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.

12. Conclusion

The worked example illustrates how to obtain Spring Stiffness Constant from measurements, graph variables and a fitted coefficient. The estimate is 19.98 N m⁻¹. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.