Period of a pendulum

This activity allows students to experimentally verify that the period of a pendulum is independent of the amplitude of the oscillations. It develops skills in precision measurement and the experimental approach in science.

In the late 16th century, a young Galileo Galilei reportedly watched a chandelier swinging in the Pisa Cathedral and noticed something remarkable: regardless of how widely it swung, the chandelier seemed to take the same amount of time for each oscillation. He timed this using his own pulse as a clock and discovered the principle of isochronism: the period of a pendulum is independent of its amplitude, at least for small swings. This property made the pendulum the most accurate timekeeping mechanism for over three centuries, from Huygens' first pendulum clock in 1656 to the quartz revolution of the 1970s. However, Galileo's observation is only approximately true. For large amplitudes, the period does increase slightly, a correction first calculated by Huygens himself. This experiment challenges students to test Galileo's claim by precisely measuring the period of a pendulum at different amplitudes and determining whether the period truly remains constant.

Activity overview:

The student designs a device to accurately measure the period of a pendulum for different release heights. By keeping the length of the pendulum constant but varying the amplitude of the oscillations, the student carries out a series of measurements then analyzes their results to check whether or not the period depends on the initial height in accordance with theoretical predictions.

Level:

Middle and high school

FizziQ

Author:

Duration (minutes) :

35

What students will do :

- Measure the period of a pendulum at several different amplitudes with high precision
- Verify Galileo's isochronism principle for small oscillations
- Detect the deviation from isochronism at large amplitudes
- Apply the formula T = 2π√(L/g) and compare theoretical and measured periods
- Develop precision measurement techniques using timing over multiple oscillations

Scientific concepts:

- Oscillation period
- Harmonic movement
- Isochronism
- Oscillation amplitude
- History of science

Sensors:

- Accelerometer (to detect oscillation timing)
- Stopwatch / Timer (manual or acoustic trigger)

Material needed:

- Newton's pendulum or simple pendulum
- Smartphone with the FizziQ application
- Stopwatch or sensor adapted to measure the period
- Tape measure for measuring heights
- Support for the pendulum
- FizziQ experience notebook

Experimental procedure:

Construct a simple pendulum using a string at least 1 meter long with a dense, compact bob (a heavy ball or a bag of coins). Attach it to a rigid support point.
Measure the pendulum length L from the support point to the center of the bob. Record this value precisely.
Calculate the theoretical period: T = 2π√(L/g), using g = 9.81 m/s².
Open FizziQ and prepare a timing method. You can use the accelerometer attached to the bob, an acoustic trigger, or a manual stopwatch.
Start with a small amplitude: pull the bob to an angle of approximately 5° (a horizontal displacement of about L × sin(5°) ≈ 0.09 × L).
Release the pendulum and time 10 complete oscillations. Divide the total time by 10 to get the period T₁. Repeat three times and average.
Increase the amplitude to 10° and repeat: time 10 oscillations, repeat three times, average.
Continue with amplitudes of 15°, 20°, 30°, 45°, and 60° (if safely achievable).
Record all data in a table: amplitude (degrees), measured period (averaged), and theoretical prediction.
Plot the measured period versus amplitude graph. For small angles (< 15°), the period should be nearly constant.
At larger amplitudes (> 30°), the period should increase noticeably. Calculate the percentage increase relative to the small-angle period.
Compare your small-angle period with the theoretical value T = 2π√(L/g). Calculate the percentage error and discuss sources of uncertainty.

Expected results:

For a 1-meter pendulum, the theoretical period is approximately 2.006 seconds. At amplitudes below 15°, the measured period should agree with this value within ±1-2% (about ±0.02 seconds). At 30°, the period increases by about 1.7%; at 45°, by about 4%; at 60°, by about 7%. These small increases may be difficult to detect without precise timing. The key to precision is timing 10 or 20 oscillations and dividing, which reduces the uncertainty by a factor of 10 or 20. Students should find that the period is approximately independent of amplitude for small swings, confirming Galileo's isochronism, but shows detectable deviations for larger swings.

Scientific questions:

- What did Galileo observe about the chandelier in Pisa, and why was it significant?
- Why is the period independent of amplitude only for small oscillations? What changes for large angles?
- How does the period depend on the length of the pendulum? On the mass of the bob?
- Why is timing 10 oscillations and dividing by 10 more accurate than timing a single oscillation?
- How did pendulum clocks achieve such high accuracy for centuries?
- On the Moon (g = 1.6 m/s²), how would the period of the same pendulum change?

Scientific explanations:

The isochronism of the small oscillations of a pendulum is one of the fundamental discoveries of Galileo in the 17th century. Contrary to intuition, the period T of a simple pendulum theoretically depends only on its length L and the acceleration of gravity g, according to the formula T = 2π√(L/g), and not on the amplitude of the oscillations.

This property is, however, only accurate for small oscillations (less than approximately 10°). For larger amplitudes, the period increases slightly according to the formula T = T₀(1 + sin²(θ/2)/16 + ...), where T₀ is the period for small oscillations and θ is the maximum angle.

This correction remains small: for an angle of 30°, the increase is only 1.7%. Accurately measuring the period can be done in several ways: 1) By manually timing several complete oscillations then dividing by their number; 2) Using FizziQ's acoustic timing to mark passages through the equilibrium position; 3) Using the accelerometer to detect acceleration maxima at the lowest point of the trajectory.

The greater the number of oscillations measured, the greater the accuracy, because the trigger errors are distributed over a greater number of periods. This property of isochronism revolutionized the measurement of time: Christiaan Huygens exploited it in 1656 to create the first precise pendulum clock, allowing time measurements with an error of only a few seconds per day.

Before this invention, the best timepieces could deviate by a quarter of an hour daily. This experiment thus makes it possible to reproduce a major scientific discovery which transformed time measurement technology and promoted advances in maritime navigation and astronomy.

Extension activities:

Frequently asked questions:

Q: My measured period does not match the theoretical value. What are the most likely causes?
R: The most common error is in measuring the pendulum length. Measure from the support point to the center of mass of the bob, not to its bottom. Also ensure the pendulum swings in a single plane without wobbling.

Q: How do I measure the amplitude angle accurately?
R: For small angles in degrees, the angle ≈ (horizontal displacement / string length) × (180/π). Measure the horizontal displacement of the bob from its rest position with a ruler.

Q: The period seems to change over time. Why?
R: Air resistance and friction at the pivot gradually reduce the amplitude. Since the period depends slightly on amplitude (for larger swings), the period may decrease slightly as the oscillations decay. This is most noticeable for large initial amplitudes.

Q: Why do I need to time 10 oscillations instead of just one?
R: The human reaction time for starting and stopping a stopwatch is about ±0.2 seconds. For a 2-second period, this is a 10% error. By timing 10 oscillations (20 seconds), the reaction time error drops to about 1%.