Period of a pendulum
Checking the isochronism of the oscillations of a pendulum
Author :
Titre 4
Learning objectives :
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.
Concepts covered
Oscillation period; Harmonic movement; Isochronism; Oscillation amplitude; History of science
What students will do :
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.
What is required :
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
Scientific background :
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.