Research design

Background

The acceleration due to gravity, g, is not the same everywhere on Earth. It is larger at the poles than at the equator, because the Earth is flattened and rotating, and it becomes smaller as you go higher above sea level. Our school laboratory is in Gaborone, at a latitude of about 24.66° S and about 1,014 m above sea level (Wikipedia, n.d.), so the value of 9.81 m s⁻² that we use in class problems may not be the true local value. I wanted to find out whether a simple pendulum, built from equipment available in our laboratory, is precise enough to measure the local value of g and to detect this kind of difference.

A simple pendulum is a small, heavy bob on a light string that swings about a fixed point. When the bob is displaced by a small angle θ, the restoring force is approximately proportional to the displacement, so the bob moves with simple harmonic motion. For small angles the period is

where L is the length from the pivot to the centre of mass of the bob (Nave, n.d.-a). The second form is the one I used: if the model is correct, a graph of T² against L is a straight line through the origin, and its gradient is 4π²g. So g can be found from the gradient of a graph that uses all nine lengths, instead of from a single measurement, and the intercept gives a check on the model and on how I measured the length. The formula is only an approximation: it is accurate for amplitudes of a few degrees, and the period becomes slightly longer as the amplitude increases (Nave, n.d.-a; n.d.-b). This is why the release angle is an important controlled variable in my method.

Variables

Choice of range, interval, repeats and equipment

I carried out a pilot trial before the main experiment. A 1.00 m pendulum had a period of about 2.0 s, and a 0.20 m pendulum about 0.9 s. At 0.10 m the period was only about 0.6 s, and I lost count of the oscillations twice in three attempts, so I chose 0.200 m as the shortest length. The longest length, 1.000 m, was set by the height of the clamp stand above the bench. Nine lengths at equal intervals of 0.100 m spread the points evenly along the graph, which is better for finding a gradient than bunching them together, and nine lengths with five repeats (45 readings) fitted into two double lessons.

Timing a single oscillation would have made my reaction time a large part of each reading. In the pilot, repeated single-swing times at 1.00 m varied by about 0.2 s, which is 10 % of the period. Timing 10 oscillations spreads the same reaction-time error over ten periods, so the percentage uncertainty in T is about ten times smaller. I did not time more than 10 oscillations because counting became unreliable at short lengths in the pilot, and the amplitude decreases noticeably over a long run. I repeated each length five times so that I could see the spread of the readings and use the half-range as the uncertainty.

The release angle was kept small because the period grows with amplitude. Using the series for a large-amplitude pendulum, T ≈ T₀(1 + θ₀²16), a 10° release (0.175 rad) makes the period only about 0.19 % longer than the small-angle value (Nave, n.d.-b). I fixed a protractor at the pivot so that I could read the angle of the thread directly, and released the bob from 8° every time.

The equation assumes that all the mass is concentrated at a point on a massless string, so I chose equipment that comes close to this. The steel bob has a mass of about 67 g, while a metre of the thin cotton thread has a mass well under 1 g. Treating a sphere of radius 12.7 mm as a point mass changes T² by less than 0.2 % even at 0.200 m, and by much less at longer lengths. I chose cotton thread instead of the nylon fishing line in the laboratory because nylon stretches more under load, which would make the length change while I was taking readings.

Method

The apparatus is shown in Figure 1.

1. Fix a clamp stand to the bench with a G-clamp. Clamp the cotton thread between the two flat plates of a pendulum clamp so that the pivot is a sharp, fixed point.
2. Measure the diameter of the steel bob in three directions with vernier calipers and record the mean (25.40 ± 0.02 mm), so the radius is 12.70 mm.
3. Adjust the thread until the distance from the bottom of the clamp plates to the top of the bob, read on a metre rule held vertically against the thread, is 0.187 m. Adding the radius (0.0127 m) gives L = 0.200 m to the nearest millimetre. Read the rule at eye level at both ends to avoid parallax (±0.001 m at each end, so ±0.002 m).
4. Pull the bob aside until the thread lines up with the 8° mark on the protractor. Release it from rest without pushing.
5. Let the bob complete two oscillations so that the swing settles. Start the stopwatch as the bob passes the lowest point of its swing, count 10 complete oscillations and stop the stopwatch when it passes the lowest point in the same direction for the tenth time. Record the time to 0.01 s.
6. Repeat steps 4–5 until there are five readings for this length. Note anything unusual, such as an elliptical swing.
7. Re-measure L to check that the thread has not slipped, then repeat steps 3–6 for L = 0.300 m to 1.000 m in steps of 0.100 m.

Safety, ethics and environment. The only real hazard was the stand tipping over, so it was clamped to the bench, and the bob was kept below head height. No chemicals or living organisms were used, so there were no ethical or disposal issues; all equipment was returned to the laboratory store.

Data analysis

Raw data

All 45 timings are given in Table 2.

* During trial 2 at L = 0.200 m the bob began to move in a slight ellipse instead of a straight line. I kept the reading because it is only 0.04 s below the lowest of the other four trials at that length (8.95 s against 8.99–9.15 s), much less than the 0.16 s spread of those four, and I had no measured reason to reject it. This was the only elliptical swing I saw; it happened at the shortest length, where the swing is smallest and quickest and a slightly sideways release has the most effect.

Processing

For each length I found the mean time for 10 oscillations and used the half-range of the five readings as its uncertainty, because the spread from reaction time (up to 0.27 s) was much larger than the 0.01 s resolution of the stopwatch. I then divided by 10 to find the period T and squared it. In Physics, the percentage uncertainty doubles when a quantity is squared. The worked example below is for L = 0.500 m.

The percentage uncertainty in T² is largest at 0.200 m (2.2 %) and smallest at 0.400 m (0.7 %). The uncertainty in L (±0.002 m) is 1.0 % at 0.200 m but only 0.2 % at 1.000 m, so I included it as horizontal error bars, although they are too short to see clearly on the graph.

Graph and gradient

Figure 2 shows T² plotted against L. I could have calculated g separately for each length and averaged the nine values. I chose a graph instead because it uses all the data at once and shows whether the relationship really is linear. It also separates a constant error in L, such as measuring to the top of the bob instead of its centre, from the gradient: such an error would only move the intercept, while it would bias every individual value of g, especially at the shortest lengths.

The points lie very close to a straight line (R² = 0.99998), with no visible curve. To find the uncertainty in the gradient, I found the steepest and shallowest straight lines that still pass through every error bar, using all nine points and not just the first and last.

Since 4π² is an exact constant, the percentage uncertainty in g is the same as the percentage uncertainty in the gradient. The y-intercept of the best-fit line is 0.003 s², and the intercepts of the maximum and minimum gradient lines are −0.045 s² and +0.046 s², so the intercept is 0.003 ± 0.046 s², which includes zero.

Conclusion

Within the range 0.200 m to 1.000 m, the square of the period of my pendulum increased linearly with its length, and the line passes through the origin within its uncertainty (intercept 0.003 ± 0.046 s²). So T² is directly proportional to L. The gradient of 4.07 ± 0.10 s² m⁻¹ gives a value for the acceleration due to gravity in Gaborone of g = 9.71 ± 0.23 m s⁻².

This matches the small-angle model for a simple pendulum. A straight line through the origin is exactly what T² = 4π²gL predicts, and the intercept is equivalent to a length of only 0.0026 s²4.07 s² m⁻¹ ≈ 0.0006 m, which is well inside my ±0.002 m uncertainty in L. This suggests that I measured the length to the correct point, the centre of the bob: if I had measured only to the top of the bob, every length would have been 12.7 mm too short, and the line would have had a positive intercept of about 4.07 × 0.0127 ≈ 0.05 s². That is only just outside my intercept range of −0.043 to +0.049 s², so my data make this error unlikely but cannot rule it out completely. This proves that the period of any simple pendulum is proportional to the square root of its length.

To judge my value of g, I needed an accepted value for Gaborone itself, not the textbook 9.81 m s⁻². The Physikalisch-Technische Bundesanstalt, Germany's national metrology institute, calculates local gravity from latitude and height using the normal gravity field of the Geodetic Reference System 1980, with a relative uncertainty of about 10⁻⁴ (PTB, n.d.). I used the same model in the form given by SensorsOne (n.d.): the International Gravity Formula 1980 with a free-air correction for height.

My value is 0.079 m s⁻² lower than this, a percentage difference of 0.80 %. This is much smaller than my percentage uncertainty of 2.4 %, so my result agrees with the accepted local value within its uncertainty. The model ignores the density of the local rock, so the true value could differ slightly, but by far less than my uncertainty. The value of 9.81 m s⁻² also lies inside my range of 9.48 to 9.94 m s⁻², so my experiment is not precise enough to show that gravity in Gaborone is weaker than the standard value; that difference (0.24 %) is ten times smaller than my uncertainty.

Although the difference is within the random uncertainty, my value is below the accepted value, and there is one systematic error that I know pushes it in that direction. Every release was at about 8°, so every period was slightly longer than the small-angle value. Using T ≈ T₀(1 + θ₀²16) (Nave, n.d.-b), an 8° amplitude increases T by 0.12 % and T² by 0.24 %, and a 10° amplitude increases T² by 0.38 %. Because this is almost the same fraction at every length, it makes the gradient too steep and g too small by the same fraction. Correcting for it would raise my value to about 9.73–9.74 m s⁻², which closes roughly a third to a half of the gap; the rest is consistent with random error in hand timing. The amplitude also decreased slightly during each set of 10 oscillations, so the real correction is probably near the lower end of this range.

Evaluation

I have ranked the weaknesses of my investigation by how much they affected my value of g, using evidence from my own data.

1. Hand timing with a stopwatch (most important; random)

Reaction time in starting and stopping the stopwatch was the largest source of uncertainty. The stopwatch reads to 0.01 s, but the half-ranges in Table 3 are between 0.045 s and 0.135 s, about five to fourteen times larger. The effect was worst at 0.200 m, where the percentage uncertainty in T was 1.1 %, compared with 0.8 % at 0.500 m and 0.5 % at 1.000 m. At this length the period is shortest and the swing is only about 28 mm each side of the rest position, so it is hardest to judge exactly when the bob passes the lowest point, and I was judging that point by eye without a marker. The elliptical swing in trial 2 made this even harder. These larger error bars at the short lengths allow the maximum and minimum gradient lines to pivot further, which is a main reason why the uncertainty in g is as large as 2.4 %.

Improvement: place a light gate at the lowest point of the swing, connected to a data logger, and record the time for 20 oscillations. This removes reaction time completely, and timing 20 oscillations halves the remaining percentage uncertainty, so lengths shorter than 0.200 m could also be used. If no light gate is available, a vertical marker behind the rest position of the bob and a mobile-phone video at 60 frames per second would allow the crossing times to be read to about 0.02 s.

2. Release amplitude (systematic; makes g too small)

As explained in the conclusion, releasing at 8° made each period 0.12 % too long, so my value of g is about 0.24 % too low (0.38 % if some releases were nearer 10°). This is a small effect compared with the 2.4 % random uncertainty, but it always acts in the same direction, so repeating the readings would not remove it. I also could not read the protractor more precisely than about ±1°, because the thread was a few millimetres in front of the scale.

Improvement: release from 5°, which reduces the effect on T² to under 0.1 %, and mount the protractor directly behind the thread so that the angle can be read without parallax. A 5° swing at 0.200 m moves only about 17 mm each side of the rest position, which is harder to time by eye, so this improvement works best together with the light gate.

3. Too few repeats to estimate the uncertainty reliably (random)

The half-range of only five readings is itself uncertain. For example, the percentage uncertainty in T² was 0.7 % at 0.400 m but 1.4 % at 0.300 m and 1.6 % at 0.500 m, the lengths on either side, even though nothing about the method changed. This makes some error bars too short and others too long, which affects how far the maximum and minimum lines can pivot.

Improvement: take ten readings at each length and use the standard deviation of the readings, or the standard error of the mean, instead of the half-range. With a light gate this would add only a few minutes per length.

4. Measuring the length (minor; random)

The ±0.002 m uncertainty in L is 1.0 % at 0.200 m but only 0.2 % at 1.000 m. The small intercept (0.003 ± 0.046 s²) is consistent with no large systematic error in where I measured the length to, although its uncertainty is equivalent to about ±11 mm of length, so an error of a few millimetres could not be detected. The random part mainly affects the shortest lengths. Improvement: measure the length of the hanging thread with a metre rule held against a set square at the clamp, and use the vernier-caliper radius for the bob as before.

5. Air resistance (minor)

Air resistance may also have slowed the bob down and affected the period. Improvement: use a heavier bob.