Simple pendulum lab report

1. Title and aim

Acceleration due to gravity with a simple pendulum. Determine gravitational acceleration from the gradient of a graph of period squared against pendulum length.

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 of 9.7982 m s⁻², differing by 0.086% 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

• Bob and light string: Concentrate the moving mass in a small bob suspended from a string whose mass and extension can be neglected.
• Rigid support and clamp: Provide a fixed pivot and allow the effective pendulum length to be changed.
• Metre rule and angle guide: Measure pivot-to-bob-centre length in metres and set an initial angle no larger than 5 degrees.
• Timer and reference marker: Measure elapsed time for 20 complete oscillations using a consistent crossing and direction.
• Bob treated as a point mass; string mass and stretching neglected.
• Fixed amplitude: 3° for the example report.
• Time is for 20 complete cycles and rounded to 0.01 s.

4. Theory and graph choice

Turn the familiar motion of a swinging bob into a quantitative measurement. You will vary length, time 20 complete oscillations at each setting, calculate period, and use a straight-line graph to estimate g. The displayed timing run is accelerated virtual time; its readings are generated by an ideal model, not observations of a physical pendulum.

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. Set a pivot-to-bob-centre length and record L in metres. Begin near the middle of the available range.
2. Displace the bob by no more than 5 degrees. Release it without a push and keep motion in a single vertical plane.
3. Choose one direction of crossing at the centre. Start at crossing zero and finish at crossing 20 in the same direction. Read the elapsed virtual time in seconds.
4. Repeat timings if available, then use at least five lengths across the permitted range. Keep the angle small for every run.
5. For each length calculate mean elapsed time, T = mean time/20, and T squared. Keep unrounded values for the fit and round only table display values.
6. Fit T squared against L with an intercept. Calculate g = 4pi squared/gradient, state units, inspect the intercept and residuals, and discuss uncertainties before comparing with the model reference.

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 = 0.0861%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 6.3516e-08; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 2.8702e-04. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 2.0030e-04.

Propagating only the relevant fit coefficient gives a fit-only standard error of 6.9799e-04 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.

Timing resolution, cycle counting, pivot-to-bob-centre length, amplitude and support motion affect a real result. Since T = t/N, timing twenty cycles reduces the fractional impact of a single endpoint timing error. An omitted finite-amplitude correction biases the inferred g downward. Length errors and correlated timing errors require additional treatment.

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

• Measure length to the bob centre at eye level to reduce parallax.
• Use a rigid support and a light, taut string; avoid rubbing at the pivot.
• Release without a push, use at most 5 degrees, and avoid elliptical motion.
• Count full cycles in the same direction and record virtual readings as simulated data.

This virtual experiment assumes a point bob, a massless inextensible string, a fixed frictionless pivot, planar small-angle motion and uniform gravity. It neglects damping, finite-bob inertia, support movement and finite-amplitude period changes. Timing displays represent accelerated simulated time rather than wall-clock measurements. Rounded model readings may create small fit residuals; these are not empirical measurement uncertainty. The reference 9.80665 m/s squared is a chosen standard model parameter, not a claim about gravity at your location. The timing model includes the leading finite-amplitude correction T ≈ 2π√(L/g)[1 + θ₀²/16 + 11θ₀⁴/3072], with θ₀ in radians. The simple graph analysis neglects this correction, giving a small systematic bias even at 3°. Angles above 5° remain available for exploration but cannot be recorded.

12. Conclusion

The worked example illustrates how to obtain acceleration due to gravity from measurements, graph variables and a fitted coefficient. The estimate is 9.7982 m s⁻². A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.