Before you start
You need these ideas and skills for this practical. If any of them are new, read this section first.
Key words
- Oscillation: one complete swing, from the starting point, across to the other side, and back to the start again.
- Period (T): the time for one complete oscillation, measured in seconds (s).
- Amplitude: how far the bob is pulled to the side, here measured as the release angle.
- Fiducial marker: a fixed reference mark (a pin or line) at the centre of the swing, used to judge exactly when to start and stop timing.
- Mean: the average of repeated readings: add them up and divide by how many there are.
Skills you need
- Measuring length with a metre rule to the nearest millimetre, with your eye level with the mark to avoid parallax error.
- Using a stopwatch and reading it to 0.01 s.
- Recording results in a table with the quantity and unit in each column heading, e.g. Length / cm.
- Calculating a mean and rounding to a sensible number of significant figures.
- Plotting a graph: a scale that uses more than half the grid, neat crosses, and a single thin best-fit line or smooth curve.
Variables
- Independent variable: the one you change on purpose. Here: the length of the pendulum.
- Dependent variable: the one you measure. Here: the period.
- Control variables: the ones you keep the same so the test is fair: mass of the bob, release angle, the string and the clamp.
Why we time many oscillations
Your reaction time when pressing a stopwatch is about 0.1–0.3 s. One oscillation of a 60 cm pendulum takes only about 1.5 s, so an error of 0.2 s is more than 10% of the reading.
If you time 10 oscillations (about 15 s) and divide by 10, the same 0.2 s error is shared between 10 swings: only about 0.02 s per period, a little over 1%. Timing 20 oscillations makes it even smaller.
Aim and apparatus
Aim
To measure the period of a simple pendulum accurately by timing several oscillations and taking a mean, and to find out how the length of the pendulum affects its period.
Syllabus link: Cambridge IGCSE Physics 0625 / 0972, topic 1.1 Physical quantities and measurement techniques: finding a short time interval by measuring multiples, including the period of a pendulum.
Apparatus
- Clamp stand with boss and clamp, fixed to the bench with a G-clamp
- Two small wooden blocks (or split cork) to grip the string
- Thread or string, and a pendulum bob (20 g, 50 g or 100 g in this lab)
- Metre rule
- Stopwatch reading to 0.01 s
- Fiducial marker (a pin or a line) placed at the lowest point of the swing
- Protractor, to keep the release angle the same
- Optional: light gate and timer instead of the stopwatch
Theory
What decides the period?
A pendulum swings at a steady rate, which is why pendulums were used in clocks for nearly 300 years.
- Length: a longer pendulum has a longer period.
- Mass of the bob: no effect. A heavy and a light bob of the same size swing with the same period.
- Amplitude: almost no effect, as long as the swing is small (below about 15°).
So length is the variable that matters, and it is the one you investigate.
Length and period are not proportional
Doubling the length does not double the period. To double the period you need about four times the length:
| Length / cm | Period / s (on Earth) |
|---|---|
| 25 | 1.00 |
| 50 | 1.42 |
| 100 | 2.01 |
That is why a graph of period against length is a curve, not a straight line through the origin.
Stretch: the pendulum equation
This goes beyond the IGCSE syllabus, but it explains the curve. For small swings:
Squaring both sides gives
so a graph of T² against L is a straight line through the origin, and its gradient lets you find g, the acceleration of free fall.
Stretch: pendulums on other worlds
The period depends on g. On the Moon (g = 1.6 N/kg) a pendulum swings much more slowly than on Earth (g = 9.8 N/kg). Use the Location buttons in the lab to test this, then use your T² graph to measure g for the Moon or Mars. This links to topic 6, Space physics.
Planning the investigation
Variables
| Type | Variable | How |
|---|---|---|
| Independent | Length L | At least 5 lengths, e.g. 20, 40, 60, 80, 100 cm |
| Dependent | Period T | Time 10 oscillations, three times; T = mean time ÷ 10 |
| Control | Mass of bob | Use the same bob every time |
| Control | Release angle | Always the same small angle, checked with the protractor (e.g. 10°) |
| Control | Number of oscillations | Always 10 |
Prediction
Write a prediction before you start, for example: “As the length of the pendulum increases, the period will increase.”
A better prediction says how: “The period will increase, but less and less for each extra 20 cm, because period and length are not proportional.”
Range and repeats
Use a wide range of lengths (at least 5 values) so that your graph shows the shape clearly. Repeat each timing three times and calculate a mean, to make the results more reliable and to spot any anomalous reading.
Method in a real lab
- Set up the stand. Fix the clamp stand to the bench with a G-clamp so it cannot move or fall over. Clamp the string between two wooden blocks so that the top of the pendulum is fixed.
- Measure the length. Let the bob hang at rest. Measure from the bottom of the wooden blocks (the pivot) to the centre of the bob, with your eye level with the scale.
- Place the fiducial marker directly below the pivot, behind the bob's lowest point.
- Release the bob. Pull it to one side by a small angle (about 10°), keep the string straight and let go without pushing, so it swings in a flat arc.
- Time 10 oscillations. Start the stopwatch as the bob passes the fiducial marker. Count “zero” as you start, then count each time it passes the marker going the same way. Stop on “ten”.
- Repeat twice more for the same length, so you have three times, t₁, t₂ and t₃.
- Change the length and repeat for at least four other lengths.
- Calculate the mean time for 10 oscillations, then the period T = mean ÷ 10, for each length.
How to use this virtual lab
- Set the length with the slider. The dashed line shows where the centre of the bob is on the metre rule.
- Choose the bob and angle. Start with the 50 g bob and 10°. The protractor at the top shows the angle.
- Press Release bob. The bob swings, and its path fades behind it.
- Time it yourself. Press Start (or the space bar) as the bob passes the red fiducial marker. The counter shows how many oscillations have passed. Press Stop as it passes the marker for the 10th time going the same way.
- Press Record reading. Your time goes into the table. Repeat the timing twice more to get t₂ and t₃; the table calculates the mean, the period T and T².
- Change the length and repeat for at least five lengths. Each length gets its own row.
- Plot the graph. Open Data & graph and press Plot graph. Look at T against L first, then T² against L.
- Test the other variables. Change the bob mass, or the angle (try 25°), and repeat a few lengths. Each set-up gets its own table, and the chips above the table switch between them.
- Compare with a light gate. Switch Timing to Light gate and press Arm gate. It times the oscillations automatically, with no reaction time. Compare its readings with yours.
- Use Slow motion if the bob is too fast to follow, and Full screen for a bigger view.
Results
Results table
Draw your table like this. The lab fills in the same columns for you. These are sample results from a real-style experiment:
| Length / cm | t₁ / s | t₂ / s | t₃ / s | Mean time for 10 oscillations / s | Period T / s |
|---|---|---|---|---|---|
| 20.0 | 8.91 | 8.85 | 9.03 | 8.93 | 0.893 |
| 40.0 | 12.53 | 12.70 | 12.64 | 12.62 | 1.262 |
| 60.0 | 15.38 | 15.54 | 15.37 | 15.43 | 1.543 |
| 80.0 | 17.92 | 17.79 | 17.80 | 17.84 | 1.784 |
| 100.0 | 20.03 | 20.18 | 19.93 | 20.05 | 2.005 |
Calculations
Example for 60.0 cm: mean = (15.38 + 15.54 + 15.37) ÷ 3 = 15.43 s, so T = 15.43 ÷ 10 = 1.543 s.
If one of the three times is very different from the other two, it is anomalous: time it again and leave it out of the mean.
Analysis and conclusion
Drawing the graph
- Put length on the x-axis (the independent variable) and period on the y-axis.
- Label each axis with the quantity and unit: Length / cm and Period / s.
- Choose scales that use more than half of the grid in each direction.
- Plot each point as a small cross, then draw a smooth best-fit curve. It should pass close to all the points, with about as many above it as below.
Conclusion
Describe the pattern and use your data: “As the length increases from 20 cm to 100 cm, the period increases from 0.89 s to 2.01 s. The graph is a curve, so the period is not proportional to the length: making the pendulum five times longer only made the period about 2.2 times longer.”
Also state what happened when you changed the mass of the bob: the period stayed the same (within experimental error), so mass does not affect the period.
Using your graph
You can read values from your curve. For example: what length of pendulum has a period of exactly 1.0 s? (About 25 cm.) What length has a period of 2.0 s, like a grandfather clock? (About 1 m.) The lab's graph tab works these out from your own data.
Stretch: finding g
Plot T² against L. The points lie on a straight line through the origin, so T² ∝ L. From the sample results the gradient is 0.0402 s²/cm = 4.02 s²/m, and
Evaluation
| Source of error | Effect | How to reduce it |
|---|---|---|
| Reaction time when starting and stopping the stopwatch | Each time is too long or too short by up to about 0.3 s | Time 10 or 20 oscillations, not one. Repeat and take a mean. Use a light gate. |
| Judging the exact moment to start and stop | Timing does not start at the same point in the swing each time | Use a fiducial marker at the centre, where the bob moves fastest, so the moment it passes is easy to see. |
| Losing count of the oscillations | The time is for 9 or 11 oscillations instead of 10 | Count “zero” as you start. Ask a partner to count aloud. |
| Measuring the length to the wrong point | Every length is wrong by the same amount (a systematic error) | Measure to the centre of the bob. Keep your eye level with the scale. |
| Releasing at different or large angles | Large swings (above about 15°) have slightly longer periods | Use a protractor and release from the same small angle every time. |
| Pushing the bob or letting it swing in a circle | The motion is not a simple swing | Release from rest, with the string straight, without pushing. |
| Top of the string slipping or the stand moving | The length changes during the experiment | Grip the string between wooden blocks and clamp the stand to the bench. |
Another variable to try: the mass of the bob, or the release angle. Investigate one at a time, keeping the length fixed.
Safety
- Clamp the stand to the bench with a G-clamp so it cannot tip over, especially with a heavy bob and a long string.
- Keep the area where the bob swings clear of people and objects.
- Do not use very large angles or swing the bob hard: it can hit someone or pull the stand over.
- Stand back from the swing and keep your face away from the path of the bob.
Exam practice
Practical-style questions like those in Paper 5 (Practical Test) and Paper 6 (Alternative to Practical). Try each one before opening the answer.
1. A student times 20 oscillations of a pendulum and gets 28.6 s. Calculate the period.
T = 28.6 ÷ 20 = 1.43 s.
2. Explain why the student timed 20 oscillations instead of just one.
One oscillation is too short to time accurately because of reaction time. Timing 20 makes the reaction-time error much smaller compared with the total time, so the period (total ÷ 20) is more accurate.
3. Suggest why timing is started as the bob passes the centre of the swing, not at the end.
The bob moves fastest at the centre, so the moment it passes a fiducial marker there is easy to judge. At the ends of the swing the bob is moving slowly and seems to pause, so the exact moment is hard to see.
4. A student says: “If I double the length, the period will double.” Use the sample results on this page to test this.
At 40 cm, T = 1.262 s; at 80 cm, T = 1.784 s. Doubling the length increased the period by a factor of 1.784 ÷ 1.262 ≈ 1.41, not 2. So the student is wrong: period is not proportional to length.
5. Name two variables that must be kept constant, and say how.
Any two of: the mass of the bob (use the same bob); the release angle (use a protractor and the same small angle each time); the number of oscillations timed (always 10).
6. The student measured the length to the top of the bob instead of its centre. What type of error is this, and how does it affect the results?
A systematic error. Every length is too short by the radius of the bob, so all the points on the graph are shifted the same amount to the left. Repeating the readings does not remove it.
7. A heavier bob of the same size is used. Predict the effect on the period.
No change. The period does not depend on the mass of the bob (for the same length and a small angle).