A simple pendulum studied with a low-cost wireless acquisition board

Abstract

In the field of physics teaching, the simple pendulum provides a very simple way to study the harmonic oscillator, source of some fundamental concepts like periodicity, relaxation time and anharmonicity. As such, its experimental study is frequently integrated in the high school or first years university programs. With this in mind, in this article we propose a simple setup allowing to study some of its properties. By coupling a small wireless acquisition board to an oscillating mass, the centripetal acceleration is recorded and analyzed. We show using basic calculations how this acceleration allow to extract the variation with time of its period and of the amplitude. The experimental results show that the damping of the amplitude is very well described by an exponential law. The accuracy we obtain on the period allows to show that, not only it decreases with amplitude, but that this decrease is relatively well described by standard calculation. Using wireless multipurpose acquisition board and usual spreadsheet for experimental modelling, the completion of this work, about 4 hours, offers to the student an original way to go a little beyond the classical study of the spectacular object that is the pendulum.

The period of the damped non-linear pendulum

“…The present measurements of the angular velocity of a pendulum for amplitudes up to 180 • were made with a MEMs gyro/accelerometer [9] mounted on the axis of the pendulum and are comparable with those of Fernandes et al [12] who measured the tangential acceleration with an off axis accelerometer for amplitudes up to 90 • , of Alho et al [18] who measured the centripetal acceleration for amplitudes up to 60 • and Pedersen [10] up to 90 • using a novel position readout with resolution of ∼0.03 • .…”

“…The question of whether damping increases or decreases the oscillation period is found to depend on the definition of the amplitude of the damped oscillation. If one takes the amplitude as its initial amplitude ϕ 0 , as is done by other authors [10,12,18], then the damped oscillation period is less than the undamped period at ϕ 0 as calculated with the AGM. However, if the undamped period is calculated with the AGM at the 'central amplitude' ϕ c , i.e.…”

“…Note that Alho et al [18] reports that the damped period is larger than the undamped period, while Fernandes et al [12] and Petersen et al [10] report that it is less. The present data, see figure 6(b), indicates that for 'central amplitudes' less than 1 radian the effects of damping on the period should be negligible, but increases the period at larger amplitudes.…”

Semiclassical calculation of the pendulum period

“…As far as its educational contribution is concerned, the simple pendulum appears first in elementary high-school physics as an illustrative example of harmonic motion with amplitude-independent period for small angles [10]. During a university course on general physics, the pendulum's swing period is calculated via elliptic integrals [11], while many experimental schemes are found for its educational demonstration [12][13][14].…”

Physical pendulum model: Fractional differential equation and memory effects

“…From the experimental point of view the study of pendular motion can be conducted by measuring the angular position of the moving object or by measuring its acceleration. Recently, different studies used the latter approach [35,54,58,59].…”