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TL;DR The simple pendulum experiment measures the acceleration due to gravity g by timing oscillations at different string lengths, then plotting T2 against L. The gradient of the best-fit line gives g=4π2/gradient
. Time 20 oscillations (not one) to cut percentage error from reaction time, and always use a fiducial marker at the equilibrium position.
What This Experiment Measures
The simple pendulum experiment determines the acceleration due to gravityg at your location. You do this by investigating how the period T of a simple pendulum depends on its length L.
This is one of the most frequently examined practicals in the O-Level Physics 6091 syllabus. It also appears as a planning question in Paper 3 - you may be asked to design the entire experiment from scratch.
Theory
For a simple pendulum swinging at a small angle (less than about 10 degrees), the period is given by:
T=2πgL
Squaring both sides:
T2=g4π2⋅L
This is in the form y=mx, where:
y=T2
x=L
m=g4π2 (the gradient)
Plotting T2 on the y-axis against L on the x-axis should produce a straight line through the origin. From the gradient, you can calculate g:
g=gradient4π2
Apparatus
You will need:
Retort stand, boss, and clamp
Inextensible string (at least 1 m long)
Small, dense bob (a metal sphere works best - steel or brass)
Metre rule (resolution 1 mm)
Stopwatch (resolution 0.01 s)
Protractor (optional, to verify the angle of release is below 10 degrees)
Fiducial marker - a pencil, thin rod, or piece of tape placed at the equilibrium (rest) position of the bob
The fiducial marker is important. It gives you a fixed reference point so you can consistently identify when the bob completes each oscillation, reducing timing uncertainty.
Step-by-Step Method
Set up the pendulum. Clamp the string firmly at the top of the retort stand so the bob hangs freely without wobbling. The pivot point should be well-defined - clamp between two flat surfaces if possible.
Measure the length L. Use the metre rule to measure from the pivot point to the centre of the bob. Record L in metres. Start with L=0.200 m (20.0 cm).
Place the fiducial marker. Position a thin vertical marker (pencil or rod) directly behind the string at the bob's rest position. This is where you will start and stop your count.
Displace the bob. Pull the bob to one side so the string makes a small angle with the vertical - no more than 10 degrees. Release it gently without pushing.
Time 20 complete oscillations. Start the stopwatch as the bob passes the fiducial marker moving in one direction. Count 20 complete oscillations (the bob returns to the same point moving in the same direction 20 times). Record the time t1.
Repeat the timing. Without changing L, repeat step 5 to get a second reading t2.
Calculate the mean and period. Find the mean time for 20 oscillations: mean time = (first reading + second reading) / 2. Then period T = mean time / 20.
Repeat for six different lengths. Use values such as 0.200, 0.300, 0.400, 0.500, 0.600, and 0.700 m. Spread them evenly across the available range for a better graph.
Why Time 20 Oscillations?
This is a point examiners specifically look for. The reason comes down to percentage error.
A typical human reaction time is about ±0.3 s. If you timed a single oscillation lasting roughly 1 s, the percentage error would be:
1.00.3×100%=30%
That is far too large. By timing 20 oscillations (total time around 20 s), the same reaction-time uncertainty gives:
200.3×100%=1.5%
This is a much more acceptable level of uncertainty. Some schools time 30 or even 50 oscillations for even greater precision.
Raw Data Table
Your raw data table should look something like this:
L / m
t1 / s (20 oscillations)
t2 / s (20 oscillations)
t(mean) / s
0.200
17.9
18.1
18.0
0.300
22.0
21.8
21.9
0.400
25.4
25.2
25.3
0.500
28.4
28.2
28.3
0.600
31.0
31.2
31.1
0.700
33.6
33.4
33.5
Note the table heading format: quantity / unit. This is the expected convention for O-Level practical work, as detailed in the Paper 3 marking guide.
Processed Data Table
Calculate T and T2 for each length:
L / m
t(mean) / s
T / s
T2 / s2
0.200
18.0
0.900
0.810
0.300
21.9
1.095
1.199
0.400
25.3
1.265
1.600
0.500
28.3
1.415
2.002
0.600
31.1
1.555
2.418
0.700
33.5
1.675
2.806
Check that T2 values are given to the correct number of significant figures - typically 3 or 4 s.f. is appropriate at this level. For more guidance on significant figures and data processing, see the practical maths toolkit.
Graph: T2 vs L
Plot T2 (y-axis) against L (x-axis). Your axes should:
Use sensible scales that fill at least half the grid in both directions
Be labelled with quantity and unit: "T2 / s2" and "L / m"
Start from the origin (0, 0) since the theory predicts the line passes through it
Draw the best-fit straight line. This should pass as close to all points as possible, with roughly equal numbers of points above and below the line. It should pass through or very near the origin. For a thorough guide to drawing best-fit lines and calculating gradients, refer to the graph skills guide.
Calculating the Gradient
Choose two points on the line (not data points) that are far apart. Draw a large triangle - examiners penalise triangles that are too small. The triangle should use at least half the length of the line.
Using the example data, suppose the two points on your best-fit line are (0.100,0.400) and (0.700,2.800):
This is close to the accepted value of 9.81 m/s², which gives confidence the experiment was carried out correctly. A small difference is expected due to experimental uncertainties.
Sources of Error and Improvements
This section is heavily tested in Paper 3. You need to identify errors and state how to reduce them. For guidance on structuring your practical write-up, see the practical report writing guide.
Reaction time when starting/stopping the stopwatch
Improvement: Time 20 (or more) oscillations rather than one, and use a fiducial marker at the equilibrium position so you can start and stop timing at a clearly defined point where the bob moves fastest.
Pendulum not swinging in a single plane
If the bob traces an elliptical path instead of swinging back and forth in a flat plane, the measured period will be inaccurate.
Improvement: Release the bob carefully from rest (do not push it). You can also set up two vertical guides (such as two retort stands) on either side of the pendulum's path to ensure it swings in one plane only.
Difficulty measuring L to the centre of the bob
The length L is defined from the pivot to the centre of mass of the bob, but you cannot place a ruler inside the bob.
Improvement: Measure the diameter of the bob using vernier calipers, then add half the diameter to the length measured from the pivot to the top of the bob. For a step-by-step guide to reading vernier calipers, see our vernier caliper and micrometer guide. Alternatively, mark the centre of the bob with a line before the experiment.
Air resistance
Air resistance causes the amplitude to decrease over time, which can introduce a small systematic error.
Improvement: Use a small, dense bob (such as a steel sphere) to minimise the surface-area-to-mass ratio. Keep the angle of swing small (below 10 degrees) to reduce the speed and therefore the drag force.
Parallax error when reading the metre rule
If your eye is not level with the marking on the rule, you will read the wrong value.
Improvement: Place your eye directly level with the measurement mark on the rule, or use a set square pressed against the rule and the string to eliminate parallax.
Paper 3 Planning Questions
The simple pendulum experiment is a classic target for the planning question on Paper 3. You may be asked something like:
"Describe an experiment to determine the acceleration due to gravity using a simple pendulum."
In your answer, you need to:
State the independent variable (length L), dependent variable (period T), and variables to be controlled (mass of bob, angle of release, same string)
Describe the setup and procedure clearly
Explain why you time 20 oscillations
State that you would plot T2 against L
Explain how to obtain g from the gradient
Identify at least two sources of error and their improvements
For the full set of Paper 3 expectations and the marking rubric, refer to the Paper 3 marking guide.
Common Exam Mistakes
Plotting T vs L instead of T2 vs L. The relationship between T and L is not linear - it is a square-root relationship. Plotting T against L gives a curve, which is much harder to analyse. Always linearise the equation first.
Not timing enough oscillations. Stating that you will time "a few" oscillations is too vague. Specify a number - 20 is the standard, and you must justify it by referencing the reduction in percentage error.
Drawing a gradient triangle that is too small. The two points you choose for the gradient calculation should be far apart on the best-fit line, using at least half the line's length. A small triangle magnifies reading errors.
Forgetting to include units in the gradient. The gradient has units because the axes have units. Here, the gradient has units of s²/m (or equivalently s² m⁻¹).
Confusing oscillation with swing. One complete oscillation is a full back-and-forth cycle - the bob leaves the fiducial marker, swings to one side, returns through the marker, swings to the other side, and returns to the marker again. One swing to one side and back is only half an oscillation.
What Next?
This experiment is part of the broader O-Level Physics practical syllabus. If you are preparing for Paper 3, work through the other core experiments and review the graph skills guide to make sure your plotting technique is exam-ready.
