Study of damped oscillations using Phyphox and Arduino controlled Hall-sensor

The paper presents physics education activities organized around the topic of damped oscillations. We used the Phyphox smartphone application for secondary school physics classes. These activities served as a basis for a physics education workshop, where an Arduino-controlled Hall-sensor and the Phyphox Magnetometer were presented. The problem of a damped pendulum, a vertical oscillation in water, and an LCr oscillating circuit was examined as part of a Phyphox project. Mechanical and electromagnetic damped oscillations can be demonstrated with our devices. Using our data, we could compare Hall-sensors of different devices, estimate some characteristics of the waves and help plan an LCr oscillating circuit. Activities for secondary school physics classes are suggested, based on the pedagogical goals.


Introduction
Sensors have more advantages when used in secondary physics classes and undergraduate course.The students benefit from increased autonomy and repeatable experiences, even at home.For the professor, it involves replacing costly devices, involving students more, and offering more complex pedagogical content.[1] In accordance with the pedagogical objective, the teacher and students have the option to use either a smartphone application or a microcontroller sensor, or both simultaneously.[2] Phyphox is a smartphone application, developed by RWTH Aachen University for physics experiences.It detects the sensors of your phone, then recommends some realizable physics experiences.[3] Remote control of Phyphox is possible through the local WiFi network, and data from the experiment can be monitored from a remote device, allowing for immediate responses when necessary.To read a more complete analysis of Phyphox application, see [4].
Using the features of Phyphox, we organized activities around the topic of damped oscillations at Berzsenyi Dániel Secondary School in Budapest (BDG) and a workshop on Innovative Methods for Physics Teachers Conference in Debrecen (IMMF).To improve the precision of measurements, we used Hall-sensors controlled by Arduino at Deutsche Schule Budapest (DSB) and at the 26th International Conference on Multimedia in Physics Teaching and Learning in Prague (MPTL'26).See Table

Damped, overdamped and critically damped oscillations
Damped, overdamped and critically damped oscillations can be described by a second-order homogeneous differential equation as (1).
The oscillation type depends on the sign of exponents of the solution of (1): (2)  1 ,  2 constants are determined by  0 ,  0 ̇ initial conditions of the system.The exponents  1 ,  2 are where and 2 0 B  = (5) Here β is the damping constant, ω 0 is the proper circular frequency of the undamped oscillatory system.The oscillation is overdamped if λ 1 > 0; λ 2 > 0, it is critically damped if λ 1 = λ 2 = 0 and it is called damped if λ 1 < 0; λ 2 < 0. [4] We focused on graphical representations of x(t) and x(t), because we try to distinguish the oscillation types from the monitor image.Notice that we have the same function type for x(t) as x(t) what we use in mechanical cases.
The solutions of overdamped and critically damped cases are formally different, however we receive the same type of graphical representation for the two cases.On Figure 1.(a) we see the graphical representation x(t) with x 0 = 0 initial condition for all the two cases.On Figure 1.(b) we represent x(t) with ẋ0 = v 0 initial condition for all the two cases.
Regarding the damped oscillation on Figure 2., with the same initial conditions as on Figure 1, this representation is visibly different to overdamped and critically damped oscillations.On Figure 2. we represented the exponential curves by dashed lines, fitted to local maxima and minima.[5][6] The proper circular frequency of damped oscillation (ω′) is different to the proper circular frequency of the undamped system (ω 0 ):

Mechanical damped oscillations
We show two mechanical examples, where we have a loss, supposedly proportional to velocity, all along the motion.In these systems we can also find a directing force, proportional and reverse direction to displacement.Here we focused only on the detection of the damped oscillation case with Phyphox.

Damped pendulum
First, we used Gyroscope sensor in Mechanics/Pendulum Phyphox box.We built a phone-holder with a toy kit to avoid the rotation of the system.To strengthen the loss, we completed it with a rigid sheet.See Figure 3.We can formulate the equation of motion of the system: where m is the mass, is the length of pendulum, g is the gravitational acceleration, x is the displacement measured from the vertical,  is called coefficient of friction.

Vertical oscillations in water
As a second mechanical damped oscillatory system, we present vertical oscillations of a smartphone in water.To follow the motion of the system we used the Accelerometer sensor in the Mechanics/Spring box.We built a smartphone holder from PVC-block, covered by a waterproof tape and we balanced it by counterbalances.We suggest to use a waterproof phone or a plastic sachet to protect your phone completely against water.See Figure 5(a).For a detailed Phyphox analysis of a pendulum, see [7].
Considering the buoyant force as directing force, we receive the same type of equation as (1): where y is the vertical displacement, measured from the equilibrium position, m is the mass, γ is the coefficient of friction, g is the gravitational acceleration.h is the total height of the PVC-block, h0 is the height of PVC-block above the water-surface in the equilibrium position.See Figure 5   First, we upload the capacitor with the battery, then we discharge it through the coil.If we formulate Kirchhoff"s loop rule of the circuit, we receive the same type of differential equation as (1): where q is the electric charge, q̇= I is the momentary current intensity in the circuit.
It indicates the existence of damped oscillations of electromagnetic waves.We tried to detect these oscillations with different devices, equipped by Hall-sensor, as in [9].For this goal, we tested Phyphox using Hall-sensor in Magnetometer box.A Hall-sensor works on the base of the charge separation mechanism of Lorentz-force, which allows to measure voltage instead of measuring magnetic induction.

Activity suggestions
If you don't know the inductance L of the coil, we suggest to determine the f 0 proper frequency of RLC-circuit (see Figure 12) with current resonance and apply Thomson formula to determine L: If you know the capacity C of capacitor, the inductance L and the resistance r of the coil, we suggest to search critically damped oscillation using (10.2) condition and check the known L value.
If you would demonstrate damped oscillations you have to satisfy first the condition (10.3).The second condition comes from the expect that we would make at least 10 measures par period.We can express it with the parameters of the circuit using (4) and updating (6) for LCr circuits: ( ) ( ) ( ) If you use a circuit according to (11), find a damped oscillation and determine the exponent β of the fitted exponential curve of absolute values of maxima and minima.As it depends only on resistance r and inductance L, you can check the measure using the relation (12).

Device comparison: period of a damped oscillation
We planned an LCr-circuit according to (13) to detect damped oscillations.It contains a capacitor of capacity 1500 F, an U-iron cored coil of 600 turns, of resistance 3.7 . and of inductance 222 mH.We tested the LCr-circuit by two different smartphones.The first smartphone measures by 20 ms, the other one by 10 ms, which was the fastest sensor in the class.See Figure 13.
We can compare the ω 0 proper circular frequency of the circuit to ω′ circular frequency of the damped oscillation using (6) and updating (5) for LCr-circuit: ( ) In the above described LCr-circuit the difference between T 0 and the damped period T' is environ 1.3 ms.Analysing Figure 13.where we have data by 10-20 ms, it is evident that smartphones measures are insufficient to detect this difference.In this case we need data, measured by less than 1 ms.
To improve the number of measures per period, we used a linear Hall-sensor (SA-15 on Figure 14(a)).The module contains a potentiometer to adjust the sensitivity of the sensor.The software requests the interval of measure, because the finite memory capacity of Arduino.Then the module fixes the reference level.If the deviation of the magnetic induction relative to the reference level is large enough, the measure starts: the deviation relative to the previous value will be recorded.The results fit to theory, especially if we use capacitors between 2000-4000 F.If you would demonstrate all types of oscillations, use a coil of inductance L between150-250 mH with a resistance lower than resistance of 5 .For demonstrations and estimations of order of magnitude we can use smartphone and Arduino controlled Hall-sensor, for more rigorous analysis we suggest to use Arduino.

Conclusion
Numerous research studies highlight the positive pedagogical impacts associated with incorporating smartphones into physics education.[1][10] In this context, we tested the Phyphox smartphone application for damped oscillations topic.Our approach involved organizing activities for more than 50 students and 30 physics teachers.Phyphox proved valuable for the demonstration of various mechanical and electromagnetic damped oscillations.For precise data analysis, within an LCr-oscillating circuit, we recommend the use of Arduino-controlled Hall-sensors.

4 .
(b).On Figure 6.(a) we can see the x(t)graphic of damped vertical oscillations and Figure 6.(b) shows the fitted curve to absolute value of acceleration maxima depending on time.We received these graphical representations after data export from Phyphox to Microsoft Excel, as in the pendulum case.See a more detailed analysis of vertical oscillations in water: [8] Electromagnetic damped oscillations: LCr oscillating circuit We call LCr oscillating circuit or LCr circuit a coil with loss, of inductance L, a resistor of resistance r and a capacitor of capacitance C. It also contains a battery and a two-way switch.See Figure 7.(a).

Figure 5 .Figure 6 .
Figure 5. (a) Vertical oscillatory system (b) Equilibrium position of the system in water

Figure 6 .
Figure 6.Vertical oscillatory system in water (a) Fitted curve to absolute values of acceleration maxima (b) x(t) graphic

4. 4 .
Different types of oscillations Figure15.resumes an LCr-circuit test.We used two coils of different turns and two arrangements (straight-iron-cored, U-iron-cored).In every arrangement we used many capacitors of different C capacity, see left columns of Figure 15.The oscillation types are marked by different colours."Calc" means theoretically calculated, see (10.1), (10.2) and (10.3)."Ard" means measured by Arduino controlled Hall-sensor, "phy" means measured by Phyphox. .

Table 1
Study of damped oscillations: activities and workshops (in bracket: number/age of participants)