Papers in Physics, vol. 10, art. 100008 (2018) www.papersinphysics.org Received: 24 July 2018, Accepted: 9 October 2018 Edited by: A. Mart´ı,M. Monteiro Licence: Creative Commons Attribution 4.0 DOI: http://dx.doi.org/10.4279/PIP.100008 ISSN 1852-4249 Physical pendulum experiment re-investigated with an accelerometer sensor C. Dauphin,1, 2∗ F. Bouquet3 We have conducted a compound pendulum experiment using Arduino and an associated two-axis accelerometer sensor as measuring device. We have shown that the use of an ac- celerometer to measure both radial and orbital accelerations of the pendulum at different positions along its axis offers the possibility of performing a more complex analysis com- pared to the usual analysis of the pendulum experiment. In this way, we have shown that this classical experiment can lead to an interesting and low-cost experiment in mechanics. I. Introduction ysis compared to the usual single measurement of the pendulum period. The physical pendulum experiment is the typical Furthermore, we use a microcontroller and an one to introduce the physics of oscillating systems. associated two axis accelerometer sensor to acquire The usual aim of the analysis of this experiment the data. Thus, we have used this simple and low is to determine the pendulum period and damping cost experiment compared to the ready to use com- factor by using an angular position sensor [1, 2]. mercial one to introduce a richer theory and data We have conducted the pendulum experiment by analysis tools that can lead to an interesting exper- using a two axis accelerometer sensor. Such sensor iment in mechanics. has already been used by Fernandes et al. (2017) [3] We have described the theoretical analysis of this but their study focused on the analysis of the time experiment and present an example of a possible ex- variation of the radial acceleration to investigate perimental setup, the analysis of the measured ra- large-angle anharmonic oscillations. dial and orbital acceleration in order to acquire the Here, we have used the accelerometer to measure moment of inertia, the center of mass, the damping both radial and orbital accelerations of the pendu- factor and the period of the pendulum. lum at different positions along its axis, which offers the possibility of performing a more complex anal- II. Example of experimental setup ∗E-mail: [email protected] We use an Arduino [4] and a two-axis accelerometer 1 Institut Villebon-Georges Charpak, Universit´eParis-Sud (B^at.490) - Rue Hector Berlioz { 91400 Orsay, France. sensor as measuring devices. The Arduino is an in- teresting choice for an experiment, as it is an easy- 2 D´epartement de Physique, University Paris-Sud, Univer- to-use and low-cost microcontroller, with a large sit´eParis-Saclay, 91405 Orsay cedex, France. user community. Even if Arduino was not initially 3 Laboratoire de Physique des Solides, CNRS, University developed as a physicist tool, it can be used in vari- Paris-Sud, Universit´eParis-Saclay, 91405 Orsay cedex, France. ous contexts of experimental physics activities (e.g., see references [5{10]). 100008-1 Papers in Physics, vol. 10, art. 100008 (2018) / C. Dauphin et al. Figure 2: Radial (red) and orbital (black) accelerations measured by the accelerometer obtained with the ex- perimental setup displayed in Fig. 1. Inset shows the Figure 1: (a) An example of an experimental setup. temporal evolution of both accelerations between 40 s (b) An accelerometer sensor attached to the centerline and 45 s. In order to have both curves on the same plot, of the pendulum, with one of its axes parallel to the orbital acceleration is shifted by a constant offset of 9 centerline. in the inset. The experimental setup that we used here is • the period of the radial acceleration oscilla- shown in Fig. 1(a). The accelerometer sensor is tions is twice that of the orbital acceleration a microelectromechanical inertial sensor which is oscillations (inset of Fig. 2). precisely calibrated by using the values +g, 0g and −g for each axis with g = 9:8 m s−2. In the next section, we will present the theory The pendulum used in this experiment is com- that explains these main features. posed of a bar on which masses can be attached to different positions. The accelerometer is attached III. Theory to the bar and positioned in such a way that one of its measurement axes lies parallel to the bar (Fig. i. Expression of the acceleration compo- 1(b)). Special care should be taken so that the nents measured by the accelerometer wires connecting the accelerometer to the board are sensor flexible enough in order not to damp the pendulum. Figure 3 shows a pendulum sketch with notations Figure 2 shows an example of data acquired by that will be used throughout this paper. O is the the accelerometer. The main features of the graph pivot point, G is the pendulum mass center and A are: the accelerometer position. L and r stands for the distance between O and G and O and A, respec- • the radial acceleration measurement decreases tively. We note I the moment of inertia of the and goes to g as t approaches the infinity. λ pendulum about the Oz axis. Applying the angular momentum theorem to the • when the radial acceleration reaches its maxi- pendulum and considering viscous damping leads mum values, the orbital acceleration is essen- to: tially equal to zero (inset of Fig. 2). ¨ _ • the radial acceleration is asymmetric about the Iλθ = −MgL sin θ − γθ (1) straight line a = g contrary to the orbital where θ is the angle between the pendulum axis acceleration that is symmetric about the line OG and the vertical axis, M is the mass of the sys- a = 0. tem and γ is the coefficient of friction. We note 100008-2 Papers in Physics, vol. 10, art. 100008 (2018) / C. Dauphin et al. small damping (κ << !) and the small angle ap- π proximation (θ0 << 2 ), which leads from Eq. (2) to: −κt θ = θ0e cos(!t) (8) _ −κt θ = −!θ0e sin(!t) (9) ¨ 2 −κt θ = −! θ0e cos(!t) (10) with the pendulum angular frequency ! = p 2 2 pg !0 − κ and !0 = =αL. Thus, the acceler- ation components measured by the accelerometer π Figure 3: Sketch of the pendulum and notations used for κ << ! and θ0 << 2 are given by: in the text. The pivot is at O, the center of mass at G and the accelerometer sensor is at A. 2 2 −2κt 2 ar = r! θ e sin (!t) 0 (11) −κt + g cos θ0e cos(!t) 2 −κt Iλ = αML where the numerical factor α depends aθ = −2κr!θ0e sin(!t) on the type of pendulum (α = 1 for a simple pen- g r (12) + − α sin θ e−κt cos(!t) dulum and α 6= 1 for a physical pendulum). We α L 0 introduce the damping factor κ = γ to obtain: 2Iλ π With θ0 << , we can further approximate these g 2 θ¨ = − sin θ − 2κθ_ (2) two expressions by: αL Forces acting on the proof mass (msensor) inside 2 2 −2κt 2 ar = r! θ e sin (!t) the accelerometer are its weight and the inertial 0 (13) 2 −2κt 2 force due to its movement. Thus, the radial and or- + g 1 − θ0e cos (!t) bital components of these forces in the non-inertial −κt aθ = −2κr!θ0e sin(!t) reference frame of the pendulum are given by: g r (14) + − α θ e−κt cos(!t) α L 0 _2 Fr = msensorrθ + msensorg cos θ (3) Some comments can be made about Eqs. (11) ¨ Fθ = −msensorrθ − msensorg sin θ (4) and (12). We first focus on the radial acceleration ar. The acceleration components, as measured by the accelerometer at a distance r from the pivot, • cos(cos(x)) and sin2(x) are both π-periodic T are then given by: functions. So ar is a 2 periodic function with 2π T = ! being the pendulum period. Indeed, the pendulum reaches its maximum velocity _2 ar = rθ + g cos θ (5) and so its maximum radial acceleration each ¨ time θ is equal to 0. aθ = −rθ − g sin θ (6) −κt • g cos (θ0e cos(!t)) varies between between Including the expression of θ¨ (Eq. (2)) into the g cos(θ0) and g (blue curve in Fig. 4(b)) and expression of aθ gives : goes to g as t approaches infinity. Physically, g r this function represents the projection of ~g a = − α sin θ + 2κrθ_ (7) θ α L onto the pendulum axis OG. _ 2 2 −2κt 2 2 2 We choose θ = θ0 and θ = 0 as initial conditions • r! θ0e sin (!t) varies between r! θ0 and of the pendulum movement. We only consider here 0 and the upper envelope of this function (red 100008-3 Papers in Physics, vol. 10, art. 100008 (2018) / C. Dauphin et al. 10.8 10.8 10.6 10.6 10.4 10.4 10.2 10.2 10 10 9.8 9.8 9.6 9.6 9.4 9.4 02 4 6 8 10 12 14 16 02 4 6 8 10 12 14 16 2 2 −2κt 2 Figure 4: (a) Calculated radial acceleration versus time. (b) In red, graph of the r! θ0e sin (!t) + g −κt ◦ contribution; in blue, graph of the g cos θ0e cos(!t) contribution. L = 30 cm, θ0 = 15 , r = 45 cm, κ = 0:1 s−1 and α = 1 for both panels. 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 02 4 6 8 10 12 14 16 02 4 6 8 10 12 14 16 g r −κt Figure 5: (a) Calculated orbital acceleration versus time.