Lab Report Sample: Determining g With a Simple Pendulum

Model Answer

Abstract

This report determines an experimental value for the acceleration due to gravity, g, using a simple pendulum. The period of oscillation was measured for six pendulum lengths between 0.20 m and 1.20 m by timing 20 complete oscillations at each length, and a graph of period squared (T²) against length (L) was plotted. As predicted by the simple pendulum equation, the graph produced a straight line through the origin, with a gradient of 4.03 s² m⁻¹. Using the relationship g = 4π² ÷ gradient, this yielded an experimental value of g = 9.80 m s⁻², within 0.1% of the accepted UK surface value of 9.81 m s⁻² and well within the estimated experimental uncertainty of ±1.3%.

Introduction

A simple pendulum consists of a small, dense bob suspended by a light, inextensible string, oscillating with small amplitude about a fixed pivot. For small angles of swing (conventionally less than about 10°), the motion approximates simple harmonic motion, and the period of oscillation, T, is related to the pendulum length, L, and the local acceleration due to gravity, g, by the standard equation T = 2π√(L/g) (Young and Freedman, 2019). Squaring both sides gives T² = (4π²/g)L, which shows that a graph of T² against L should be a straight line passing through the origin, with gradient equal to 4π²/g. This linear relationship is more useful experimentally than the original square-root form, because a straight-line graph allows a gradient to be determined from many data points at once, averaging out random error far more effectively than a calculation of g from a single length and period (Hughes and Hase, 2010).

Historically, pendulum timing was one of the earliest precise methods for determining g and remains a standard undergraduate technique because it requires only simple, low-cost apparatus while illustrating core experimental physics skills: taking repeated readings, applying a linearisation technique to a non-linear relationship, and propagating measurement uncertainty through to a final result (Bevington and Robinson, 2003). An alternative, and in principle more direct, approach would be to calculate g separately for each individual length using g = 4π²L/T² and then average the six resulting values. This single-point method is generally considered less reliable than the gradient method used here, however, because any systematic error affecting one particular length – for example, a slightly imprecise length measurement at a single point – feeds directly into that one calculated value of g, whereas a straight-line fit through several points spreads the influence of any one anomalous reading across the whole data set and is therefore less sensitive to error in any single measurement (Squires, 2001; Taylor, 1997). For this reason, the gradient method was adopted as the primary means of extracting g in this investigation, with the aim of measuring the period of a simple pendulum across a range of lengths, using the resulting T²–L graph to obtain an experimental value of g, and evaluating the precision of that value against the accepted value of 9.81 m s⁻².

Method

Apparatus: a 20 g brass pendulum bob, thin inextensible string (approximately 1.3 m), a clamp stand with a split cork to fix the pivot point, a metre ruler (±1 mm), a stopwatch (±0.01 s, reaction time estimated separately), and a protractor for setting the initial release angle.

Procedure: the string was clamped at the pivot and the length, L, measured from the point of suspension to the centre of the bob using the metre ruler, for six lengths: 0.20, 0.40, 0.60, 0.80, 1.00 and 1.20 m. At each length, the bob was displaced through a small angle, estimated with the protractor to be less than 10° from the vertical, and released from rest. Once the oscillation had settled into a steady, regular swing, the stopwatch was started as the bob passed through the equilibrium (lowest) point, and the time for 20 complete oscillations was recorded; timing 20 oscillations rather than one substantially reduces the percentage effect of human reaction time on the final period. This was repeated three times at each length, and the mean time for 20 oscillations was used to calculate the mean period, T = mean time ÷ 20.

Controlled variables: the same bob and string were used throughout to keep mass and string elasticity constant; the release angle was kept below 10° at every length to remain within the small-angle approximation underlying T = 2π√(L/g); and air resistance was minimised by using a dense, streamlined bob and allowing the pendulum to settle into a smooth swing (free of sideways wobble) before timing began. The clamp stand was checked to ensure it was firmly secured to the bench and did not vibrate or shift during oscillation, since any movement of the pivot point would introduce an additional, uncontrolled source of error into the period measurement. A fixed reference marker was placed at the equilibrium position of the bob to give a consistent visual cue for starting and stopping the stopwatch at exactly the same point in each swing.

Risk assessment: the main hazards identified were the swinging bob causing minor injury if it struck a person or knocked equipment from the bench, and the clamp stand toppling if not adequately secured. These were mitigated by keeping a clear working area around the pendulum, ensuring the clamp stand base was weighted and stable, and standing back from the swing path while timing.

Results

Table 1 shows the mean time for 20 oscillations, the calculated period T, and T² for each of the six pendulum lengths tested.

Table 1. Mean time for 20 oscillations, period T and T² at six pendulum lengths (n = 3 per length).

Figure 1 shows T² plotted against L. The points lie close to a straight line passing near the origin, consistent with the relationship T² = (4π²/g)L predicted for simple harmonic motion at small amplitude.

Figure 1. Period squared (T²) against pendulum length (L), with gradient used to calculate g.

Worked calculation. The gradient of the line was found using the first and last data points on the fitted line:

gradient = (4.834 − 0.804) ÷ (1.20 − 0.20) = 4.030 ÷ 1.00 = 4.03 s² m⁻¹

Rearranging T² = (4π²/g)L to make g the subject gives g = 4π² ÷ gradient. Substituting the gradient found above:

g = 4 × π² ÷ 4.03 = 39.48 ÷ 4.03 = 9.80 m s⁻²

Comparing this with the accepted value of 9.81 m s⁻², the percentage difference was:

(|9.80 − 9.81| ÷ 9.81) × 100 = 0.10%

Uncertainty analysis. The length was measured to ±1 mm with the metre ruler, giving a percentage uncertainty at the shortest length (0.20 m) of (0.001 ÷ 0.20) × 100 = 0.5%. Reaction time when starting and stopping the stopwatch was estimated at ±0.2 s per timing; over 20 oscillations, this gives a percentage uncertainty in the time of (0.2 ÷ 17.9) × 100 ≈ 1.1% at the shortest length, falling to (0.2 ÷ 44.0) × 100 ≈ 0.5% at the longest length. Since the period is time divided by 20, and T² involves squaring, the percentage uncertainty in T² is approximately twice the percentage uncertainty in T. Combining the length and timing uncertainties in quadrature gives an overall estimated uncertainty in g of approximately ±1.3%, or ±0.13 m s⁻², so the final result is reported as g = 9.80 ± 0.13 m s⁻².

Discussion

The experimental value of g = 9.80 ± 0.13 m s⁻² agrees with the accepted UK surface value of 9.81 m s⁻² well within the estimated uncertainty, and the percentage difference of only 0.10% indicates that the method used – timing 20 oscillations at each of six lengths and extracting g from the gradient of a linearised T²–L graph – was both accurate and appropriately precise for an undergraduate-level investigation. The strong linearity of the T²–L graph, with all six points lying close to a single straight line through the origin, provides good evidence that the small-angle approximation held throughout and that the simple pendulum model was appropriate across the full range of lengths tested (Young and Freedman, 2019).

The uncertainty analysis showed that timing precision, rather than length measurement, was the larger source of uncertainty at the shorter pendulum lengths, where a fixed ±0.2 s reaction-time uncertainty represents a proportionally larger fraction of the shorter total oscillation time. This is consistent with general good practice in pendulum experiments, where longer lengths (and therefore longer periods) are generally preferred, since they reduce the relative impact of a fixed reaction-time uncertainty on the final period (Hughes and Hase, 2010). Using an automated timing method, such as a light gate positioned at the equilibrium point of the swing, would remove human reaction time from the measurement almost entirely and could reduce the overall uncertainty in g by a substantial margin.

Several further limitations should be noted. First, the pendulum length was measured from the pivot to the centre of the bob using a hand-held ruler, which introduces some uncertainty in locating the exact centre of a spherical bob, particularly for the shorter lengths where this represents a larger fraction of the total length. Second, air resistance was not accounted for explicitly; although the bob was dense and the amplitude small, air resistance and any residual sideways or elliptical motion of the bob (rather than a pure planar swing) would tend to make the measured period slightly longer than the ideal simple-pendulum prediction, which could contribute to the small (though not statistically significant) shortfall of the calculated g below the accepted value. Third, the assumption that the string is perfectly light and inextensible is only an approximation; a string with measurable mass would behave more like a physical pendulum with a slightly different effective length, though this effect is expected to be negligible for the light string used here.

Overall, the closeness of the calculated gradient-based value of g to the accepted value, combined with a clearly linear T²–L relationship across all six lengths, indicates that systematic errors in this experiment were small relative to the estimated random uncertainty. Extending the range of lengths tested, increasing the number of oscillations timed at each length, or replacing manual timing with a light-gate sensor would each be expected to further reduce the uncertainty in future repeats of this investigation.

It is also worth reflecting on why the gradient method proved effective here. Because every one of the six lengths contributed a data point to the same straight-line fit, small random errors at any individual length – for instance, a slightly early or late stopwatch press on one particular trial – had only a limited effect on the overall gradient, since the fitted line was constrained to pass close to all six points simultaneously rather than being determined by any single measurement. This averaging effect is the main statistical advantage of linearising a non-linear relationship before extracting a physical constant, and it explains why the final uncertainty in g (±1.3%) was smaller than the percentage uncertainty in the individual period measurements at the shortest length (approximately 2.2% in T², before averaging across the full data set). This principle – that fitting a trend across many measurements typically yields a more reliable result than relying on any one measurement in isolation – is a general feature of experimental physics that extends well beyond this particular pendulum investigation.

Conclusion

Using a simple pendulum and a linearised T²–L graph, this investigation obtained an experimental value of g = 9.80 ± 0.13 m s⁻², in close agreement with the accepted value of 9.81 m s⁻² and within the estimated experimental uncertainty of ±1.3%. Timing precision at shorter pendulum lengths was identified as the dominant source of uncertainty, and the use of longer pendulum lengths or an automated timing method such as a light gate is recommended to improve precision in any future repeat of this experiment.

References

• Young, H.D. and Freedman, R.A. (2019) University Physics with Modern Physics. 15th edn. Harlow: Pearson.
• Hughes, I.G. and Hase, T.P.A. (2010) Measurements and Their Uncertainties. Oxford: Oxford University Press.
• Bevington, P.R. and Robinson, D.K. (2003) Data Reduction and Error Analysis for the Physical Sciences. 3rd edn. New York: McGraw-Hill.
• Taylor, J.R. (1997) An Introduction to Error Analysis. 2nd edn. Sausalito: University Science Books.
• Serway, R.A. and Jewett, J.W. (2018) Physics for Scientists and Engineers. 10th edn. Boston: Cengage Learning.
• Nave, C.R. (2016) ‘Simple pendulum’, HyperPhysics. Atlanta: Georgia State University.
• Halliday, D., Resnick, R. and Walker, J. (2013) Fundamentals of Physics. 10th edn. Hoboken: Wiley.
• Squires, G.L. (2001) Practical Physics. 4th edn. Cambridge: Cambridge University Press.
• Institute of Physics (2014) Practical Physics: Measurement and Uncertainty. London: Institute of Physics.