Implementing smartphones with their internal sensors into physics experiments represents a modern, attractive, and authentic approach to improve students’ conceptual understanding of physics. In such experiments, smartphones often serve as objects with physical properties and as digital measurement devices to record, display, and analyze quantities such as the angular velocity, linear acceleration, magnetic flux, sound pressures, light intensity, etc. For example, the MEMS accelerometer and gyroscope are utilized to study the dependence of the radial acceleration on the angular velocity in circular motions and oscillation periods or the acceleration due to gravity via different pendulum setups.

Implementing smartphones with their internal sensors into physics experiments represents a modern, attractive, and authentic approach to improve students’ conceptual understanding of physics.^{1–3} In such experiments, smartphones often serve as objects with physical properties and as digital measurement devices to record, display, and analyze quantities such as the angular velocity, linear acceleration, magnetic flux, sound pressures, light intensity, etc.^{4–6} For example, the MEMS accelerometer and gyroscope are utilized to study the dependence of the radial acceleration on the angular velocity in circular motions^{2},^{7–9} and oscillation periods^{10,11} or the acceleration due to gravity via different pendulum setups.^{12–14}

The motions of different types of pendula and elastic properties of materials belong to the traditional and rather challenging topics treated in experimental physics courses at the beginning of a science education at universities. In this paper we propose an experiment in which the smartphone is used to measure the shear modulus of a material via a torsion pendulum. The smartphone with its orientation dependent moments of inertia^{4} serves as pendulum bob. While it performs torsional oscillation on a long thin wire, its MEMS gyroscope records the rotational oscillation, allowing us to determine the angular frequency and oscillation period with high accuracy. For recording the data in our torsion pendulum experiment, we chose the application phyphox^{15} (RWTH Aachen, Germany), available for Android and iOS smartphones.

## Theoretical background

For the experimental determination of the shear modulus of a material, a torsion pendulum with a pendulum bob of known moment of inertia *I*_{i} attached to a long thin wire can be used. The equation of motion for this torsion pendulum is^{16–19}

where *φ*(*t*) denotes the angle of rotation and *κ* is the directional moment of the wire. Equation (1) follows from Newton’s second law for rotational motions. It describes the angular acceleration $\phi \xa8(t)$ of the pendulum bob caused by the torque of the twisted wire, which under the assumption of Hooke’s law is proportional to −*κ* · *φ*(*t*) (for more details see e.g. Ref. 17).

A special solution of the differential Eq. (1) is the rotational oscillation

where *φ*_{0} is the initial amplitude of the angle *φ* at time *t* = 0. The parameters *ω*_{0} and *T* denote the angular frequency and the oscillation period of the undamped rotational oscillation^{16}

For the calculation of the moments of inertia *I*_{i} (*i* = *x*, *y*, *z*) of the pendulum bob, the smartphone is approximated as a homogeneous cuboid^{4} (see Fig. 1). For the rotation around the *y*-axis, one finds^{16,19}

where *a* and *c* are the lengths of the two edges of the cuboid oriented perpendicular to the *y*-axis. *I*_{x} and *I*_{z} are determined with the corresponding perpendicular edge lengths. The three *I*_{i} values represent the principal moments of inertia of the cuboid.

With the assistance of Eqs. (3) and (4) we calculate the directional moment of the torsion wire^{19}

Its shear modulus *G* (the elastic property we are heading for) is found by^{19}

where *l* and *r* denote the length and the radius of the wire, respectively. The proportionality between the directional moment of the thin wire and the shear modulus of the wire material expressed in the left part of Eq. (6) follows from an integration of the shear stress over the cross section of the wire. For further explanations regarding the shear modulus and its connection to the directional moment, see Refs. 17–19.

## Experimental setup

The torsion pendulum (see Fig. 1) is constructed with a copper wire with a radius of *r* = (0.200 ± 0.001) mm and a length of *l* = (1.22 ± 0.01) m. The torsion wire must be fixed to the stand material (not visible in Fig. 1 due to the length of the wire) and to the smartphone in a way that the rotation of the smartphone always only results in a twisting of the wire. We realized this by attaching the metal clamp via the VELCRO® fastener wrapped tightly around the smartphone.

The smartphone (for dimensions and mass see Table I) is attached to this wire in a way that the axis of rotation corresponds to one of its axes of symmetry. The rotational oscillation is initiated by carefully twisting the smartphone out of its equilibrium position. Generally, we used an initial amplitude of the angle of rotation of about $270\xb0\phi 0\u22433\pi 2$ to obey Hooke’s law for shear stress and to ensure reproducible experimental conditions.

length a (x-direction) | (7.81 ± 0.01) cm |

height b (-direction) | (15.84 ± 0.01) cm |

width c (z-direction) | (0.75 ± 0.01) cm |

mass m | (239 ± 1) g |

length a (x-direction) | (7.81 ± 0.01) cm |

height b (-direction) | (15.84 ± 0.01) cm |

width c (z-direction) | (0.75 ± 0.01) cm |

mass m | (239 ± 1) g |

The MEMS gyroscope of the smartphone running the application phyphox^{15} measures the angular velocity *ω*(*t*) instead of the angle of rotation *φ*(*t*). This is sufficient for our experiment since the angular velocity *ω*(*t*), given by

oscillates with the same angular frequency *ω*_{0} as the angle of rotation. Note that the position of the MEMS gyroscope within the smartphone is not relevant to the experiment, because the angular velocity, which is recorded, is invariant of translational shifts. In other words, *ω* is the same for all points within a rotating ridged body, regardless of the distance from the axis of rotation.

The remote-control function of phyphox connects the smartphone during the experiment to a second device (see Fig. 1). This setup allows us to record and transfer the *ω*(*t*) data in real time without interfering with the motion of the torsion pendulum.

## Results and discussion

As an example of the measured data, Fig. 2 displays the torsional oscillation *ω*_{y}(*t*) of the smartphone around its internal *y*-axis. The data show a damped rotational oscillation. Obviously, friction causes a decay of the amplitude as illustrated by the exponential decaying function ± *A*· exp(−*γ* ·*t*), which are shown by the green lines in Fig. 2. Friction also causes a slight increase of the period of the damped oscillation *T*_{d} compared to *T*. Therefore, the data were fitted with the following equation,

to determine *T*_{d} and the damping coefficient *γ*. Table II shows the results for the three different oscillations including the coefficients of determination *R*^{2} characterizing the quality of the fits. The measurement uncertainties for *T*_{d} obtained via the fit procedure are in the order of 10^{−3} s. However, phyphox records the data of the MEMS gyroscope with a time resolution of 10^{−2} s. Therefore, we used this value as an estimate for the maximum measurement uncertainties of the oscillation periods (see Table II).

axis i
. | γ [s^{−1}]
. | T_{d}[s]
. | R^{2}
. |
---|---|---|---|

x | 0.011 ± 0.001 | 14.58 ± 0.01 | 0.9956 |

y | 0.014 ± 0.001 | 7.41 ± 0.01 | 0.9991 |

z | 0.010 ± 0.001 | 16.25 ± 0.01 | 0.9981 |

axis i
. | γ [s^{−1}]
. | T_{d}[s]
. | R^{2}
. |
---|---|---|---|

x | 0.011 ± 0.001 | 14.58 ± 0.01 | 0.9956 |

y | 0.014 ± 0.001 | 7.41 ± 0.01 | 0.9991 |

z | 0.010 ± 0.001 | 16.25 ± 0.01 | 0.9981 |

The period of the undamped oscillation *T* is calculated from *T*_{d} and *γ* with^{18,19}

Table III summarizes the oscillation periods *T* of the different rotations of the smartphone around its three principal axes and the shear moduli *G* resulting by application of Eq. (6). It also includes the principal moments of inertia of the smartphone *I*_{i} calculated by using the data in Table I.

axis i
. | I_{i} (with i = x,y,z) [kg · m^{2}]
. | T [s]
. | G [GPa]
. |
---|---|---|---|

x | (5.008 ± 0.013)·10^{−4} | 14.59 ± 0.02 | 45.2 ± 1.5 |

y | (1.226 ± 0.011)·10^{−4} | 7.41 ± 0.02 | 42.8 ± 1.7 |

z | (6.212 ± 0.017)·10^{−4} | 16.24 ± 0.02 | 45.1 ± 1.8 |

axis i
. | I_{i} (with i = x,y,z) [kg · m^{2}]
. | T [s]
. | G [GPa]
. |
---|---|---|---|

x | (5.008 ± 0.013)·10^{−4} | 14.59 ± 0.02 | 45.2 ± 1.5 |

y | (1.226 ± 0.011)·10^{−4} | 7.41 ± 0.02 | 42.8 ± 1.7 |

z | (6.212 ± 0.017)·10^{−4} | 16.24 ± 0.02 | 45.1 ± 1.8 |

The uncertainties for the shear moduli *u*(*G*) given in Table III result from the estimation considering the measurement uncertainties of the radius *u*(*r*) and the length *u*(*l*) of the wire, and the oscillation period *u*(*T*) and the moments of inertia *u*(*I*) of the smartphone:

Within these uncertainties, the three reported values for the shear modulus of the copper wire agree with each other. The averaged value of $G\xaf=(46.5\xb12.0)$ GPa falls into the range of tabulated values for copper (*G* ranges from 38 GPa to 47 GPa, see e.g. Refs. 18, 21, and 22). We conclude that shear moduli of thin wires may be measured with sufficient accuracy via the proposed torsion pendulum experiment using the smartphone as pendulum bob and as digital measurement device.

## Final remarks

Our physics teacher trainees perform this experiment in groups of two as homework during their first experimental physics course. Our goal is a sustainable development and improvement of their scientific competences by joining the rather complex topics of rotational motion of solid bodies and elastic properties of materials with an attractive experiment that the students may perform, analyze, and understand under their own control and at their own speed.

A comparison of the *T*_{d} and *T* data in Tables II and III shows that the damping of the rotational oscillation of the smartphone only slightly increases the oscillation periods, although the damping is clearly observable in the enveloping function of the measurement data. We consider this result as an important practical experience for our students. It leads also to a simplification of the data analysis, e.g., for physics courses in secondary schools or less demanding tasks for physics teacher trainees at universities. If the determined damping coefficient *γ* is sufficiently small (i.e., $\gamma \u226a2\pi T$ ) it is possible to measure the oscillation period of the torsional oscillation by reading the time required for a certain number of completed oscillations directly from the recorded (*t*) data.For example, if we read from Fig. 2 the time required for six complete oscillations, we obtain a value of (44 ± 1) s. The estimated oscillation period is (7.3 ± 0.2) s, resulting in a shear modulus of *G* = (44.1 ± 2.6) GPa, which agrees with the values in Table III. However, this increases the measurement uncertainty of the oscillation period, raising the uncertainty of the shear modulus.

The experiment could be realized with torsion wires from different materials as well. One needs to consider the change of the oscillation period, which results from the change of the shear modulus. We consider oscillation periods from 2 s up to 20 s to be optimal for data acquisition. Shorter periods may lead to insufficient time resolution while longer periods result in unpractical measurement times and large data sets. As can be seen in the rewritten Eq. (6), we can estimate the oscillation period via:

For a steel wire where the shear modulus is approximately three times the shear modulus of copper, a much smaller radius or a larger length of the wire would be necessary, which might be difficult to realize for students’ experimental work task but which could of course be used in demonstration experiments in lectures. We performed a comparable experiment with nylon threads with lengths ranging from *l* = 0.5 m to 1.4 m, and radii *r* of 0.35 and 0.5 m. These experiments with nylon threads also worked very well and yielded torsion moduli between 0.7 GPa and 1.1 GPa. However, there are no consistent reference values for the shear modulus of nylon. We assume that *G* depends e.g. on the degree of polymerization and on the thread production procedure of the manufacture.

## Acknowledgment

The authors are grateful for financial support received via the STIL project of the University of Leipzig, which is supported by the Federal Ministry of Research and Education (BMBF) of Germany, Grant Number 01PL16088. We also thank the phyphox development team at RWTH Aachen (Germany) for discussion and support.

## References

**Andreas Kaps** *studied mathematics and physics for the teaching profession at high schools. Since 2019, he is a PhD student at the Department of Physics Didactics at the University of Leipzig (Germany). He works on the development and the didactic analysis of smartphone-based experimental learning environments.*

**Tobias Splith,** *PhD, is a research associate in the Department of Physics Didactics at the University of Leipzig. His research is focused on the development of smartphone-based experimental exercises and the students’ conceptual understanding of data analysis.*

**Frank Stallmach,** *PhD, with postdoctoral lecturing qualification, is lecturer for Experimental Physics at the University of Leipzig (Germany). He worked as post doctoral fellow in the USA and scientist in Norway in the field of magnetic resonance applications for material sciences before he became a research associate and lecturer at Leipzig University. Currently, he leads the university didactics subgroup at the Department of Physics Didactics. His research focuses on the creation of digital teaching and learning environments for the education of future physics teachers and physicists.*