Gravity by Coincidence Pendulum lab report

1. Title and aim

The Acceleration of Gravity by the Coincidence Method (Simple Pendulum & Master Clock). Determine the local acceleration due to gravity g to high metrological precision by measuring the coincidence beat intervals between a variable-length simple pendulum and a standard seconds pendulum.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Acceleration due to gravity g of 9.8068 m s⁻², differing by 0.002% 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 Variable-Length Simple Pendulum: Comprising a dense turned brass spherical bob suspended by a fine unspun invar wire from a hardened agate knife-edge bearing, allowing strictly planar oscillation without torsional twisting or lateral wobble.
• Standard Master Seconds Clock Pendulum: A temperature-compensated invar master pendulum maintaining a reference period of T₀ = 2.00000 s (frequency f₀ = 0.50000 Hz) with an illuminated vertical index mark.
• Optical Slit Coincidence Viewing Telescope: Equipped with crosshairs and aligned with the common vertical equilibrium plane of both pendulums to observe the exact instant of transit coincidence without parallax error.
• Precision Digital Lap-Timer / Chronometer: Resolving to 0.01 s to register the precise epochs of successive transit coincidences across multiple coincidence periods.
• Precision Vertical Cathetometer / Steel Comparator Scale: Equipped with a vernier scale resolving to 0.05 mm to measure the true vertical distance from the knife-edge bearing axis to the center of mass of the spherical bob.
• Rigid Anti-Vibration Wall-Bracket Stand: A massive cast-iron wall mount bolted to a load-bearing laboratory pillar to eliminate virtual lengthening caused by support stand compliance.
• Master seconds pendulum calibrated period T₀ = 2.00000 s (half-period = 1.00000 s)
• Test pendulum spherical brass bob diameter d_bob = 2.80 cm (radius r = 1.40 cm)
• Suspension: high-tensile invar wire with agate knife-edge bearing to suppress flexure
• Angular release amplitude θ₀ ≤ 2.5° to satisfy circular arc small-angle limit (cos θ ≈ 1 − θ²/2)
• Optical coincidence fiducial telescope aligned with equilibrium central vertical index
• Standard atmospheric condition corrections: ambient air buoyancy and Stokes drag accounted

4. Theory and graph choice

The coincidence method, pioneered by Jean-Charles de Borda, Pierre-Simon Laplace, and Friedrich Bessel, represents one of the most elegant metrological techniques in experimental physics, serving as a temporal analogue to the mechanical Vernier caliper. When measuring the period T of an oscillating pendulum with an ordinary manual stopwatch, human reaction time uncertainty (typically ±0.15 to ±0.25 s) limits precision, even over 50 or 100 oscillations. In the coincidence method, the test pendulum oscillates alongside a standard master seconds pendulum whose period T₀ is calibrated to 2.00000 s (half-period = 1.00000 s). When both pendulums pass through their central equilibrium vertical fiducial in the same direction simultaneously, a coincidence occurs. Because the period T of the test pendulum differs slightly from T₀, the faster pendulum progressively leads the slower one until, after a large coincidence time interval t_c, it has gained exactly one complete oscillation cycle relative to the master clock. Over this interval, the master clock executes N₀ = t_c / T₀ cycles while the test pendulum executes N = N₀ + 1 = t_c / T cycles. Dividing the timing interval by the elapsed cycles decouples human reaction latency, reducing measurement uncertainty to less than one part in ten thousand (0.01%). By systematically varying the pendulum length L and determining T from coincidence epochs, plotting T² against L yields a straight line with slope S = 4π² / g, from which the acceleration due to gravity g is derived with rigorous metrological accuracy.

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 m⁻¹.

5. Method

1. Mount the test pendulum on the rigid wall-bracket stand and align the viewing telescope crosshair with the vertical equilibrium index mark shared by both the test pendulum and the master seconds pendulum.
2. Using the precision cathetometer, measure the total suspended length L from the knife-edge pivot line to the top of the spherical bob, add the bob radius r = d/2, and record the effective length L to the nearest 0.1 mm.
3. Set the master seconds pendulum into steady oscillation. Displace the test pendulum through a small angle (θ ≤ 2.5°) in the strictly planar direction of swing and release it smoothly from rest without lateral wobble.
4. Observe through the telescope as both pendulums oscillate. Note the initial transit coincidence where both wires simultaneously cross the crosshair traveling in the same direction (e.g. left to right), and start the chronometer.
5. Count the intervening swings and record the chronometer reading at the second, third, and fourth consecutive coincidences traveling in the same direction. Compute the mean coincidence interval t_c.
6. Calculate the high-precision period T = (T₀ · t_c) / (t_c + T₀). Repeat the procedure across five distinct pendulum lengths L from 70.0 cm to 90.0 cm, construct a plot of T² versus L, and determine the acceleration of gravity g from the regression slope.

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 = 0.0017%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 6.9865e-08; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 9.6516e-04. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 7.7514e-04.

Propagating only the relevant fit coefficient gives a fit-only standard error of 0.00235 m s⁻². This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Optical crosshair coincidence transit detection latency (±0.20 s over multi-cycle interval t_c, scaled down by factor (T/t_c)² ≈ 0.0005 to ~0.0001 s in period T), cathetometer vernier length indexing error (±0.1 mm from knife-edge to bob center of mass), small circular arc amplitude decay (θ ≤ 2.5°, circular arc correction ~0.012%), knife-edge friction and agate plane compliance, ambient air buoyancy reducing apparent gravitational weight by ~0.014%, and virtual hydrodynamic added fluid mass contribute experimental uncertainty. The OLS regression standard error evaluates random scatter across length stations, but does not capture unmodelled buoyancy corrections or knife-edge wear.

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

• Release the test pendulum smoothly from rest using a mechanical release or thread burn technique to avoid imparting lateral or elliptical swinging modes.
• Keep the angular oscillation amplitude small (θ₀ ≤ 2.5°) throughout all timing runs to satisfy the small-angle simple harmonic approximation and avoid non-linear Duffing softening.
• Align the optical viewing telescope strictly perpendicular to the plane of swing to prevent optical parallax when identifying coincidence transits across the crosshair.
• Ensure the laboratory doors and windows are closed to eliminate air currents and thermal drafts that could disturb the laminar boundary layer around the suspended wire and bob.

The teaching model assumes an idealized planar simple pendulum with knife-edge suspension. In a physical laboratory, small corrections for the finite radius of the spherical bob, the non-zero mass of the suspension wire, ambient air buoyancy, and support stand compliance must be applied in ultra-high precision gravimetry. The coincidence method provides an extraordinary illustration of phase vernier metrology in experimental physics. By measuring the coincidence interval between the test pendulum and a standard master seconds clock, the period T is determined with a precision of one part in ten thousand, enabling an exceptionally accurate determination of the local acceleration due to gravity g.

12. Conclusion

The worked example illustrates how to obtain Acceleration due to gravity g from measurements, graph variables and a fitted coefficient. The estimate is 9.8068 m s⁻². A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.