Does a Pendulum’s Period Depend on Its Amplitude?
Activity 1 | Week 1 | Experiment
“In the fields of observation, chance favours only the prepared minds.”
– Louis Pasteur
1 The question
A common claim is that the period of a simple pendulum does not depend on its amplitude. Your task is to test this claim using a real pendulum by comparing its period at starting angles of 10\(^\circ\) and 20\(^\circ\).
Because no two measurements will be exactly the same, your conclusion must consider both:
- the difference between the mean periods; and
- the uncertainty associated with the measurements.
This reflects an important principle: a measured value becomes scientifically meaningful only when its uncertainty is considered alongside it1.
2 Learning focus
2.1 Not as simple as it might seem
Designing a convincing experiment requires many choices, even when the system being studied (a simple pendulum in case) appears simple.
Today’s activity has two goals:
- to reach a reasoned conclusion about whether the two periods are distinguishable; and
- to explain how your experimental decisions, measurements, and evidence led to that conclusion.
The second goal is more important for your future. Without conscious attention, it is easy to follow a procedure without noticing that you are making important choices about measurement design, data quality, and the strength of the evidence2. The hallmark of a good scientist, and a good problem-solver, is the ability to identify a problem, propose a sensible response, and explain why that response is justified by evidence. This activity therefore includes deliberate pauses to examine those choices.
Your central task is to recognise the experimental degrees of freedom, even those that appear trivial, and make deliberate, scientifically informed decisions.
For example:
- What counts as one complete oscillation?
- How many oscillations should be timed?
- How many repeated measurements should be made?
- What should remain constant?
- Which imperfections can safely be ignored for the present question?
- Could any assumption affect your ability to answer the question?
When scientific principles and the available evidence leave more than one reasonable option, judgement is necessary. That judgement should be conscious and supported by clear reasoning.
Please also bear in mind that an experiment is rarely a one-off activity. Instead, it is an iterative process of planning, testing, evaluating, and revising (Figure 1).
flowchart TB
A("**Plan or Revise**") --> B("**Test**")
B --> C("**Evaluate**")
C -.->|**Iterate**| A
By the end of the activity, your group should be able to:
- Plan a measurement by explaining how your design addresses the question.
- Monitor the experiment by identifying problems and making justified changes.
- Evaluate the evidence by comparing the measured difference with its uncertainty.
2.2 Working with others
Make space for everyone’s thinking
Most scientific work, especially interdisciplinary science, is a social activity. Good collaboration involves more than dividing up the work. It depends on how group members listen, question, disagree, and respond to one another.
Productive teamwork includes:
- acknowledging each contribution;
- giving every member opportunities to contribute to planning, measurement, and analysis;
- asking questions about evidence and reasoning;
- treating disagreement as a way to examine an idea rather than as personal criticism; and
- checking that the group understands a decision before acting on it.
In this sense, some useful discussion starters include:
- “What evidence supports that idea?”
- “Shall we quickly test that assumption?”
- “I see it differently because…”
- “Can we hear what others think?”
- “What should we change, and why?”
Developing these skills do not happen automatically just because you work in a group. It requires deliberate practice on your part.
Learn from other groups
You are surrounded by other groups doing the same experiment. Talk to them!
Other groups may have made different choices, noticed different problems, or found useful ways to improve the measurement. After your group has recorded its own plan and prediction, compare ideas with one or two other groups.
For example, ask:
- What did they define as one complete oscillation?
- How many oscillations did they plan to time?
- How did they set the starting angles?
- What problems did they anticipate?
- What would they do if their measurements had a large spread?
3 The Activity
3.1 Step 1 | Plan
Quickly explore
Before making a sensible measurement plan, you need to understand the system and the equipment available to you.
For now, simply set up the pendulum and let it swing. Observe how the system behaves, identify the parameters that might affect its motion, and notice the choices you make while setting it up. Do not begin taking formal measurements yet; we are just warming up.
Make a prediction
Before collecting data, briefly record your responses to the following questions:
- Do you expect the period at \(20^\circ\) to be the same as, larger than, or smaller than the period at \(10^\circ\)?
- What size of difference would be meaningful in this experiment?
- Identify at least three features of the experiment that could make the comparison less reliable. Classify each as a source of random uncertainty, systematic bias, or invalid data.
This prediction is also part of your scientific warm-up. It is entirely acceptable for the experimental result to disagree with it, and it is also acceptable not to know the answers yet. The purpose is to make your initial thinking visible so that you can revisit it later.
Design
As a group, discuss and record the following decisions.
| Decision | Group choice | Reason |
|---|---|---|
| What counts as one complete oscillation? | ||
| How many oscillations will be timed in each trial? | ||
| How many repeated measurements will be made? | ||
| How will the 10\(^\circ\) and 20\(^\circ\) angles be set? | ||
| What reference point will be used to start and stop the timer? | ||
| What will be done if the measurements have a large spread? |
For example, the definition of one oscillation must be precise enough for another group to reproduce it. One oscillation could be defined as the time taken for the bob to return to the same position and direction of motion.
Try to keep the following conditions as constant as possible within your experiments:
- bob and support
- release procedure
- timing method
- number of oscillations timed
- timing reference point and
- lighting and surroundings.
Do not begin the main experiment until the group can explain its choices.
3.2 Step 2 | Run a pilot experiment
Now run a small pilot experiment.
The purpose of the pilot is to develop a more thorough practical understanding of the procedure and to setup an experimental pipeline. Use it to identify difficulties in:
- setting the angle;
- releasing the bob;
- counting oscillations;
- operating the timer; and
- recording observations consistently.
The release procedure should be consistent:
- Pull the bob to the marked angle.
- Hold the bob still.
- Release it without flicking, pushing, or adding sideways motion.
Use the pilot data to set up a simple Python analysis pipeline on Colab that can:
- convert total measured time into a period;
- calculate summary statistics; and
- produce a quick visualisation of the results.
The focus is on the experiment and your scientific decisions—not on producing a fancy program.
Pilot measurements should normally remain separate from final measurements if the procedure or design changed.
3.3 Step 3 | Collect data
Using what you learned from the pilot, collect data for starting amplitudes of \(10^\circ\) and \(20^\circ\).
Record every measurement, including those that appear inconvenient or unusual. Do not silently delete a result because it does not fit your expectations. Also record anything that might have affected the measurement.
A typical data table might look like this:
| Amplitude | Attempt | Timed by | Released by | Number of oscillations | Total time (s) | Observations |
|---|---|---|---|---|---|---|
| \(10^\circ\) | 1 | |||||
| \(10^\circ\) | 2 | |||||
| \(10^\circ\) | 3 | |||||
| \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | ||
| \(20^\circ\) | 1 | |||||
| \(20^\circ\) | 2 | |||||
| \(20^\circ\) | 3 | |||||
| \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) |
3.4 Step 4 | Check the data
Pause and inspect your measurements before moving to the final analysis.
Ask:
- Do the numbers make physical sense?
- Is the spread of the measurements reasonable?
- Are any measurements unusual?
- If so, do your observations explain why?
- Are the uncertainties small enough to answer the original question?
Compare your data with those collected by other groups. Differences do not necessarily mean that one group is wrong. Look for differences in apparatus, technique, and experimental decisions that might explain the results.
3.5 Step 5 | Analyse and conclude
Basic calculations
For each amplitude, calculate:
- the mean period;
- the standard deviation of the repeated periods; and
- the standard uncertainty in the mean.
For \(N\) repeated measurements,
\[ s_{\overline{x}}=\frac{s_x}{\sqrt{N}}, \]
where \(s_x\) is the standard deviation of your data.
Once we have done this, there are several options for arriving at a conclusion. Some of these options are shown below. Work through all of them and decide which is the most robust.
Option 1: Calculate just the difference
Calculate the difference between the mean periods:
\[ \Delta T=\overline{T}_{20}-\overline{T}_{10}. \]
A positive value means that the measured period at \(20^\circ\) is longer.
Is this conclusive?
Option 2: Compare the numerical results
Compare the two mean periods along with their standard errors.
Do not decide only by comparing the standard deviations. The uncertainty in each mean is more relevant to the comparison.
Do the measurements provide clear evidence of a difference in period?
Option 3: Make a visual comparison
Plot the mean period for each amplitude, with uncertainty bars showing the standard uncertainty in the mean.
Use the plot to examine the pattern in the data. However, overlapping uncertainty bars alone should not determine your conclusion.
Does the plot provide clear evidence of a difference in period?
Option 4: Calculate a confidence interval for the difference
Calculate a 95% confidence interval for \(\Delta T\).
Use the interval to guide your conclusion:
- If the interval includes zero, the experiment does not clearly resolve a difference between the periods.
- If the interval excludes zero, the measurements support a difference between the periods.
- If the data are affected by substantial random uncertainty, systematic bias, or invalid measurements, explain why better measurements may be needed before drawing a firm conclusion.
3.6 What is your conclusion
Answer the following questions:
- Which statement best describes the evidence?
- The periods are distinguishable with the present measurement precision.
- The periods are not distinguishable with the present measurement precision.
- The comparison is inconclusive, and better measurements are needed.
- What numerical evidence supports your conclusion?
- Which aspect of the experiment most limits your confidence?