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. Explore a virtual apparatus, collect readings and analyse a graph.
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.
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.
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.
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.
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.
Resolving to 0.01 s to register the precise epochs of successive transit coincidences across multiple coincidence periods.
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.
A massive cast-iron wall mount bolted to a load-bearing laboratory pillar to eliminate virtual lengthening caused by support stand compliance.
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. Suggested independent settings: 70.0, 75.0, 80.0, 85.0, 90.0 cm.
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.
Precise optical alignment at the equilibrium line ensures that transit coincidences are observed at the exact moment of maximum bob velocity, minimizing trigger timing uncertainty.
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.
The true pendulum length must extend from the rotational axis to the center of mass of the sphere; adding the radius r prevents systematic underestimation of L.
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.
Restricting amplitude to ≤ 2.5° keeps circular arc period elongation below 0.012% and prevents coupling into lateral or torsional swinging modes.
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.
Triggering on simultaneous transits in the identical direction of swing defines phase difference zero (Δφ = 0 mod 2π), initiating the coincidence timing interval.
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.
Averaging across multiple coincidence epochs divides human stopwatch trigger latency by the large number of elapsed oscillation cycles, boosting precision by two orders of magnitude.
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.
The linear regression of T² against L isolates g = 4π²/S, eliminating any constant systematic length offset and providing rigorous metrological validation.
Points are your observations. The fit uses SI x and y values before display scaling.
| # | Pendulum Length L (cm) | Coincidence Interval t_c (s) | Pendulum length L (m) | Period squared T² (s²) | Remove |
|---|
No readings yet. Take a virtual measurement and record it.