Pendulum period
01 // Question
How does the length of a pendulum's string change the time it takes to complete one swing?
One full swing out and back is called a period, written as T.
02 // Variables
Independent
Length of the pendulum string- 20 cm
- 40 cm
- 60 cm
- 80 cm
Dependent
Period of one full swing- Time for 10 swings, seconds
- Period T = time ÷ 10
03 // Hypothesis
IF the length of a pendulum's string increases, THEN its period will increase, BECAUSE a longer pendulum has farther to travel on each swing, and its period depends on the square root of its length.
04 // Procedure
1. Set the length
Measure the string from the pivot to the center of the mass, starting at 20 cm.
2. Pull to a small angle
Pull the bob back about 5° or less from straight down, then release without pushing.
3. Time 10 swings
Start and stop as the bob passes the same reference point in the same direction; time 10 complete cycles.
4. Find the period
Divide the total time by 10 to get the period of one swing.
5. Repeat at each length
Repeat 3 trials at each length, then increase the length and repeat.
Materials // Prepare the bench
- String
- small bob or washer
- retort stand or secure support
- meter stick
- stopwatch
- tape
- eye protection if required
05 // Data table
| Length (cm) | Time for 10 swings — trials 1 / 2 / 3 (s) | Mean period T (s) | T² (s²) |
|---|---|---|---|
| 20 | ____ / ____ / ____ | ____ | ____ |
| 40 | ____ / ____ / ____ | ____ | ____ |
| 60 | ____ / ____ / ____ | ____ | ____ |
| 80 | ____ / ____ / ____ | ____ | ____ |
Formula T = 2π √(L ÷ g), where L is length in meters and g ≈ 9.8 m/s²; for a T² vs L graph, slope = 4π² ÷ g
06 // What to expect
The period gets longer as the string gets longer, but not in a straight line — doubling the length does not double the period. Plotting T² against L instead of T against L straightens the curve into a line, because T² is directly proportional to L.
In the ideal small-angle model, changing bob mass does not change the period. Real measurements can differ slightly because of timing scatter, air resistance, pivot friction, and differences in bob shape or size.
07 // Lab notes
- Keep the release angle about 5° or less when comparing length. Larger angles make the small-angle approximation less accurate.
- Time 10 swings instead of 1 — small timing errors on a single swing become a much bigger percentage of your result.
- Air resistance and friction at the pivot are the main real-world sources of error, and both slowly shrink the swing over time.
08 // Remember
The ideal model omits bob mass. For small swings, length and gravity set the ideal period.
09 // Measure a pendulum length consistently
Pendulum length is measured from the pivot point to the center of the bob, not just the exposed string. Mark several lengths on one string so the pivot and bob stay comparable across trials. Keep the release angle small and use the same release method each time; pushing the bob adds energy and can change its motion. Make sure the support cannot tip and that the swinging path is clear.
Time ten or twenty complete oscillations rather than a single swing. Start and stop the timing at the same reference point in the cycle, then divide total time by the number of swings to estimate one period. Repeat each length at least three times and average the period. Counting many swings reduces the effect of human reaction time on one start-stop measurement. Record the number of swings so the calculation can be checked.
10 // Why graph period squared against length
For a simple pendulum at small angles, the period depends approximately on the square root of its length. Squaring the measured period gives a quantity that is expected to vary roughly in direct proportion to length. Plot length on the horizontal axis and period squared on the vertical axis. A near-straight trend is stronger evidence for the model than merely saying that a longer pendulum 'looks slower.' The relationship is approximate and depends on the small-angle condition.
Keep the bob mass fixed when testing length. In the ideal model, mass does not determine the period, but changing the bob can also change shape and air resistance, creating an unnecessary source of variation. Report length in meters and period in seconds; the squared-period unit is seconds squared. If the graph curves or one point sits far from the trend, check measurement, release angle, length-to-center measurement, and whether the support moved.
11 // Reduce timing error and extend the test
Choose a small swing angle and a clear zero-crossing marker. One partner releases without pushing, one counts, and one times; switch roles and repeat to see whether one observer is consistently early or late. Do not stop the bob by grabbing the string while it is moving. If the string slips at the clamp, the effective length changes during a run and the data no longer represent the intended setting.
An extension can compare two small release angles while length stays fixed, or investigate whether bob mass changes period under the same setup. Make a prediction before testing and specify which factor is changing. Use a scatterplot with a best-fit line rather than connecting every point in sequence. Conclusions should distinguish the observed classroom range from a universal rule: the ideal equation assumes a small angle, a light inextensible string, a point-like bob, and negligible air resistance and friction.
12 // Turn timing into a period you can compare
A period is one complete back-and-forth cycle: the bob returns past the same reference point moving in the same direction. Time ten cycles for each of three trials, divide each total by ten, and calculate the mean period while keeping all three readings visible. This reduces the percentage effect of stopwatch reaction time without hiding trial-to-trial scatter. Keep the release angle about 5° or less and release without a push. Measure length from the pivot to the bob's center; measuring only the exposed string creates a systematic error.
For a small-angle ideal pendulum with a fixed pivot and negligible drag, T = 2π√(L/g); bob mass cancels from this model. Larger angles, air resistance, pivot friction, and a bob that is not compact can shift real measurements. Plot T against L to see a curve, then plot T² against L: the ideal line has slope 4π²/g, so g = 4π² ÷ slope. Convert length to meters before using the formula, label graph axes with units, and describe scatter instead of forcing every point onto a perfect line.
13 // Questions people ask
Does bob mass affect the pendulum period?
In the ideal simple-pendulum model at small angles, mass cancels from the period equation, so the same length and local gravity predict the same period for any bob mass. In a classroom setup, air resistance, bob shape, pivot friction, and timing precision can create small differences.
Why time 10 swings instead of just 1?
Any small error in starting or stopping the stopwatch is the same size no matter how many swings you time, but it becomes a much smaller percentage of the total time when spread across 10 swings instead of 1.
How can I time the period more reliably?
Time ten or twenty complete swings and divide the total by the number of swings. Start and stop at the same point in the motion, such as when the bob passes the center moving in the same direction. Repeat the timing and average the results to reduce the effect of stopwatch reaction time.
14 // Keep going
Can you explain the investigation?
Pendulum period · Student Data Sheet
Question: How does the length of a pendulum's string change the time it takes to complete one swing?
Hypothesis: If the length of a pendulum's string increases, then its period will increase, because a longer pendulum has farther to travel on each swing, and its period depends on the square root of its length.
Materials
- String
- small bob or washer
- retort stand or secure support
- meter stick
- stopwatch
- tape
- eye protection if required
Safety before starting
- Clamp or hold the support securely and use a light bob. Keep faces, fingers, and other people outside the swing path; do not wrap the string around a hand or neck.
- Have an adult help with any elevated support, and stop if the stand shifts or the bob becomes loose.
Lab notes
- Keep the release angle about 5° or less when comparing length. Larger angles make the small-angle approximation less accurate.
- Time 10 swings instead of 1 — small timing errors on a single swing become a much bigger percentage of your result.
- Air resistance and friction at the pivot are the main real-world sources of error, and both slowly shrink the swing over time.
Variables
Independent: Length of the pendulum string · Dependent: Period of one full swing
Keep constant: same bob · same release angle (about 5° or less) · same fixed pivot and stand
Procedure
- Set the length: Measure the string from the pivot to the center of the mass, starting at 20 cm.
- Pull to a small angle: Pull the bob back about 5° or less from straight down, then release without pushing.
- Time 10 swings: Start and stop as the bob passes the same reference point in the same direction; time 10 complete cycles.
- Find the period: Divide the total time by 10 to get the period of one swing.
- Repeat at each length: Repeat 3 trials at each length, then increase the length and repeat.
Record your observations
| Length (cm) | Time for 10 swings — trials 1 / 2 / 3 (s) | Mean period T (s) | T² (s²) |
|---|---|---|---|
Formula: T = 2π √(L ÷ g), where L is length in meters and g ≈ 9.8 m/s²; for a T² vs L graph, slope = 4π² ÷ g
Answer key and expected pattern
The period gets longer as the string gets longer, but not in a straight line — doubling the length does not double the period. Plotting T² against L instead of T against L straightens the curve into a line, because T² is directly proportional to L.
In the ideal small-angle model, changing bob mass does not change the period. Real measurements can differ slightly because of timing scatter, air resistance, pivot friction, and differences in bob shape or size.
Remember: The ideal model omits bob mass. For small swings, length and gravity set the ideal period.