New measurements of a simple pendulum using acceleration sensors
NNew measurements of a simple pendulum using accelerationsensors.
Julien Vandermarli`ere
Saint-Exupery Secondary School, 13 avenue Paul Alduy,66100 Perpignan, France; [email protected]
Mikhail Indenbom
Department of Physics, University of West Brittany,6 avenue Le Gorgeu, 29200 Brest, France; [email protected]
To measure oscillation of a simple pendulum was probably a first idea coming to mind af-ter appearance of smartphones with small but powerful acceleration sensors : Simply attachthe telephone to a playground swing or hang it on two string as the pendulum bob andrecord the data. But immediately the problem becomes not so trivial. To deeply investigateon it or to make the phase diagram of the movement require complex calculations that arefar beyond the capability of the youngest students. In this article we propose another wayto study the pendulum by putting the sensors at the axis of rotation. This method givesimmediately the figures very close to ones in textbooks. THEORETICAL BACKGROUND
For a simple pendulum, where all mass is concentrated in its bob, the tangential accel-eration L ¨ θ is caused by the component of the gravitational force in this direction − mg sin θ ( m is the pendulum mass, L is its length and θ , ˙ θ , ¨ θ are the deviation angle and its first andsecond derivatives by time, correspondingly). So, one would expect to measure this accel-eration using an accelerometer put on the pendulum bob. But the accelerometer measuresnot the acceleration vector (cid:126)a but the vector (cid:126)a − (cid:126)g ( (cid:126)g is the gravitational acceleration of afree fall pointing vertically down). Thus the measured value in the direction of motion x iszero : A x = L ¨ θ + g sin θ = 0 . (1) a r X i v : . [ phy s i c s . e d - ph ] M a y Additional forces of friction and a more complicated mass distribution (physical pendulum)give some, usually small, deviations of A x from zero. The indication of the accelerometer along the pendulum length is created by the cen-tripetal acceleration and the axial component of gravitation : A y = L ˙ θ + g cos θ . (2)Considering small oscillations θ = θ cos( ωt ) and taking into account cos θ ≈ − θ we getthat A y oscillates near g with double frequency and an amplitude proportional to θ . Thusthe signal should disappear very fast with reduction of θ . All these facts make measurementsof A y not very appropriate for characterization of the pendulum motion.If one wants to use the accelerometer for its original purpose in smartphones : to de-termine the orientation of it relatively to vertical (cid:126)g , he should place the probe so that itdoes not move and only rotates, namely, in our case, at the pendulum rotation axis . In thisposition the accelerometer will indicate a x = g sin θ and a y = g cos θ ( y is always pointing upalong the pendulum) giving direct information on the rotation angle θ . For small deviationsof the pendulum from the vertical θ ≈ a x /g simply. THE EXPERIMENT
In order to illustrate the above statements we have created a pendulum where the smart-phone can be placed in two positions: down as the pendulum bob, and up so, that thelocation of the accelerometer coinsides with the rotation axis (Fig. 1). For being moreclear we have used an additional SensorTag (CC2650STK of Texas Instruments) in order tohave measurements in two positions at the same time.The parallel recordings of two sensors were realized using phyphox software. . Usingthis software and numerous examples given in free access one can create his own complicatedexperiments with different sensors at the same time and easily save obtained data for furthertreatment, by means of Python in our case.The results presented here are obtained with the smartphone in the down (usual) positionand the SensorTag in the upper position at the rotation axis as can be seen in Fig. 1. Thesmartphone accelerometer demonstrates ordinary curves (upper panel in Fig. 2) : nearlyno signal in x direction ( A x ≈
0) and A y oscillating around g with the double frequency FIG. 1. Pendulum with two points of measurements. (1) : A smartphone used as the pendulumbob which measures ordinary effects. (2) : A SensorTag measuring at the same time the pendulumdeviation θ and its rotation speed ˙ θ . A colorful helix (3) was added to increase the dumping ofoscillations. of the pendulum. One can see how at small amplitudes it is sinking in the noise while theoscillations can be seeing still very well.The accelerometer at the rotation axis (central panel in Fig. 2) shows, as expected, largeoscillations from which the deviation angle can be obtained directly : θ = arcsin( a x /g ). Theadvantage of such measurements particularly becomes evident in the presence of the pendu-lum dumping specially introduced to the system to compare large and small amplitudes inthe same figure. A cc e l e r a t i o n a t ( m / s ) A x A y A cc e l e r a t i o n a t ( m / s ) a x a y time (s) R o t a t i o n ( r a d / s ) ω z FIG. 2. Measurements by the acceleration sensors at different points. Upper panel : The sensor(here of a smartphone) is in the usual position at the pendulum bob. A x measured in the motiondirection drops to zero immediately after releasing the pendulum ( t ≈ a x follows well the oscillationsof the deviation angle θ . Lower panel: Pendulum rotation speed ˙ θ (gyroscope component ω z perpendicular to the oscillation plane) measured at the same time. To complete this experiment we have added the gyroscope of the SensorTag to our pro-gram (lower panel in Fig. 2) and obtained ( θ, ˙ θ ) phase diagram (Fig. 3). It shows well how, θ (rad) ˙ θ = ω z ( r a d / s ) FIG. 3. Phase portrait of the pendulum oscillation. Red line outlines the initial motion by handfrom the equilibrium position to the release point. after being put to the initial position (red line in the figure) and released, the pendulumfollows clockwise the spiral of the damped oscillations (blue line). In order to see this betterthe damping was increased by adding a helix to the pendulum. Our ( θ, ˙ θ ) phase diagramis that we find in textbooks unlike ( ˙ θ, ¨ θ ) obtained by Monteiro et al. One can see how therotation speed ˙ θ ( z component measured by the gyroscope ω z ) has the maximum when θ = 0and vice verso, all as should be for harmonic oscillations of a simple pendulum. CONCLUSION
In conclusion, we should point out that the proposed measurements with sensors atthe rotation axis give much more clear and direct information on the pendulum motion.It is also not worse to mention the higher sensitivity for small oscillations clearly seen.Moreover, the comparison of the measurements between the two positions makes it possibleto highlight surprising counterintuitive phenomena like zero signal observed in the motiondirection ( A x = 0) or the double frequency oscillations of the signal A y along the pendulum. Ann-Marie Pendrill and Gary Williams, “Swings and slides,” Phys. Educ. , 527–533 (2005). Patrik Vogt and Jochen Kuhn, “Analyzing simple pendulum phenomena with a smartphoneacceleration sensor,” Phys. Teach. , 439–440 (2012). Joo C Fernandes, Pedro J Sebastio, Lus N Gonalves and Antnio Ferraz, “Study of large-angleanharmonic oscillations of a physical pendulum using an acceleration sensor,” Eur. J. Phys. ,045004 (2017). Martn Monteiro, Cecilia Cabeza and Arturo C Mart, “Exploring phase space using smartphoneacceleration and rotation sensors simultaneously,” Eur. J. Phys. , 045013 (2014). Sidney Mau, Francesco Insulla, Elliot E. Pickens, Zihao Ding and Scott C. Dudley, “Locating asmartphone’s accelerometer,” Phys. Teach. , 246–247 (2016). Chris Isaac Larnder, “A Purely Geometrical Method of Locating a Smartphone Accelerometer,”Phys. Teach. , 52–54 (2020). Christoph Stampfer, Heidrun Heinke and Sebastian Staacks, “A lab in the pocket,” Nat. Rev.Mater. , 169–170 (2020). phyphox (physical phone experiments) Web Site,