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 motions2,7–9 and oscillation periods10,11 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 inertia4 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 phyphox15 (RWTH Aachen, Germany), available for Android and iOS smartphones.

For the experimental determination of the shear modulus of a material, a torsion pendulum with a pendulum bob of known moment of inertia Ii attached to a long thin wire can be used. The equation of motion for this torsion pendulum is16–19 

φ¨(t)+κIiφ(t)=0,
(1)

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 φ¨(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

φ(t)=φ0sinω0t+π2,
(2)

where φ0 is the initial amplitude of the angle φ at time t = 0. The parameters ω0 and T denote the angular frequency and the oscillation period of the undamped rotational oscillation16 

ω0=κIiT=2πω0.
(3)

For the calculation of the moments of inertia Ii (i = x, y, z) of the pendulum bob, the smartphone is approximated as a homogeneous cuboid4 (see Fig. 1). For the rotation around the y-axis, one finds16,19

Fig. 1.

Picture of the torsion pendulum to measure the shear modulus. The smartphone is attached via VELCRO® fasteners and a metal clamp to the long thin torsion wire, which is not shown in full length.

Fig. 1.

Picture of the torsion pendulum to measure the shear modulus. The smartphone is attached via VELCRO® fasteners and a metal clamp to the long thin torsion wire, which is not shown in full length.

Close modal
Iy=112m(a2+c2),
(4)

where a and c are the lengths of the two edges of the cuboid oriented perpendicular to the y-axis. Ix and Iz are determined with the corresponding perpendicular edge lengths. The three Ii 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 wire19 

κ=4π2IiT2.
(5)

Its shear modulus G (the elastic property we are heading for) is found by19 

G=2lκπr4=8πlIir4T2,
(6)

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.

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°φ03π2 to obey Hooke’s law for shear stress and to ensure reproducible experimental conditions.

Table I.

Dimensions and mass of the smartphone as given by the manufacturer.20 

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 phyphox15 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

ω(t)=φ˙(t)=φ0ω0cosω0t+π2,
(7)

oscillates with the same angular frequency ω0 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.

As an example of the measured data, Fig. 2 displays the torsional oscillation ωy(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 Td compared to T. Therefore, the data were fitted with the following equation,

Fig. 2.

Angular velocity ωy(t) of the smartphone oscillating around its internal y-axis on the torsion wire. The black circles represent the measured data. The red line shows the fit according to Eq. (8). The green lines illustrate the enveloping function of the dampened oscillation (see text).

Fig. 2.

Angular velocity ωy(t) of the smartphone oscillating around its internal y-axis on the torsion wire. The black circles represent the measured data. The red line shows the fit according to Eq. (8). The green lines illustrate the enveloping function of the dampened oscillation (see text).

Close modal
ω(t)=Aexp(γt)sin2πtt0Td,
(8)

to determine Td and the damping coefficient γ. Table II shows the results for the three different oscillations including the coefficients of determination R2 characterizing the quality of the fits. The measurement uncertainties for Td 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).

Table II.

Damping coefficient γ, period of oscillation of the damped motion Td, and coefficient of determination R2 as determined by the fit procedure using Eq. (8).

axis iγ [s−1]Td[s]R2
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 iγ [s−1]Td[s]R2
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 Td and γ with18,19

T=2π2πTd2+γ2.
(9)

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 Ii calculated by using the data in Table I.

Table III.

Principal moments of inertia Ii of the smartphone, the corresponding oscillation periods T calculated with Eq. (9), and shear moduli G of the copper wire obtained from the torsion pendulum experiments.

axis iIi (with i = x,y,z) [kg · m2]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 iIi (with i = x,y,z) [kg · m2]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:

u(G)=Glu(l)+Gru(r)+GTu(T)+GIu(I).
(10)

Within these uncertainties, the three reported values for the shear modulus of the copper wire agree with each other. The averaged value of G¯=(46.5±2.0) 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.

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 Td 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., γ2πT ) 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:

T1r2lG.
(11)

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.

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.

1.
J.
Briggle
, “
Analysis of pendulum period with an iPod touch/iPhone
,”
Phys. Educ.
48
,
285
288
(
May
2013
).
2.
M.
Monteiro
,
C.
Cabeza
, and
A.
Marti
, “
Rotational energy in a physical pendulum
,”
Phys. Teach.
52
,
312
313
(
May
2014
).
3.
J.
Chevrier
,
L.
Mandani
,
S.
Ledenmat
, and
Ahmad
Bsiesy
, “
Teaching classical mechanics using smartphones
,”
Phys. Teach.
51
,
375
376
(
Sept.
2013
).
4.
M.
Patrinopoulos
and
C.
Kefalis
, “
Angular velocity direct measurement and moment of inertia calculation of a rigid body using a smartphone
,”
Phys. Teach.
53
,
564
565
(
Dec.
2015
).
5.
K.
Hochberg
,
J.
Kuhn
, and
A.
Müller
, “
Using smartphones as experimental tools - Effects on interest, curiosity and learning in physics education
,”
J. Sci. Educ. Technol.
27
,
385
403
(
April
2018
).
6.
J.
Kuhn
and
P.
Vogt
, “
Smartphone & Co. in Physics Education: Effects of Learning with New Media Experimental Tools in Acoustics
,” in
W.
Schnotz
,
A.
Kauertz
,
H.
Ludwig
, A. Müller, and
J.
Pretsch
(Eds.),
Multidisciplinary Research on Teaching and Learning
(
Palgrave Macmillan
,
Basingstone, UK
,
2015
).
7.
K.
Hochberg
,
S.
Gröber
,
J.
Kuhn
, and
A.
Müller
, “
The spinning disc: Studying radial acceleration and its damping process with smartphone acceleration sensors
,”
Phys. Educ.
49
,
137
140
(
March
2014
).
8.
A.
Shakur
and
T.
Sinatra
, “
Angular moment
,”
Phys. Teach.
51
,
564
565
(
Dec.
2013
).
9.
A.
Kaps
and
F.
Stallmach
, “
Tilting motion and the moment of inertia of the smartphone
,”
Phys. Teach.
58
,
214
215
(
March
2020
).
10.
J.
Kuhn
and
P.
Vogt
, “
Analyzing spring pendulum with a smartphone acceleration sensor
,”
Phys. Teach.
50
,
439
440
(
Oct.
2012
).
11.
J. C.
Palacio
,
L.
Velazquez-Abad
, and
Gimenez J.A.
Monsorio
, “
Using a mobile phone acceleration sensor in physics experiments on free and damped harmonic oscillations
,”
Am. J. Phys.
81
,
472
475
(
June
2013
).
12.
J.
Kuhn
and
P.
Vogt
, “
Analyzing radial acceleration with a smartphone acceleration sensor
,”
Phys. Teach.
51
,
182
183
(
March
2013
).
13.
W.
Wong
,
T.
Chao
,
P.
Chen
,
Y.
Lien
, and
C.
Wu
, “
Pendulum experiments with three modern electronic devices and a modeling tool
,”
J. Comp. Educ.
2
,
77
92
(
Feb.
2015
).
14.
U.
Pili
, “
Measurement of g using a magnetic pendulum and a smartphone magnetometer
,”
Phys. Teach.
56
,
258
259
(
April
2018
).
15.
The phyphox homepage by the RTWH Aachen, https://phyphox.org.
16.
D.
Halliday
,
R.
Resnick
, and
J.
Walker
,
Fundamentals of Physics
, 6th ed. (
Wiley Inc
.,
2010
), p.
441
.
17.
P.
Tipler
and
G.
Mosca
,
Physics for Scientists and Engineers,
6th ed. (
Houndmills
,
Basingstone, UK, New York
,
2008
), pp.
474
475
.
18.
D.
Giancoli
,
Physics for Scientist and Engineers,
4th ed. (
Pearson Education
,
London
,
2009
), pp.
321
and 383–384.
19.
W.
Demtröder
,
Mechanics and Thermodynamics – Undergraduate Lecture Notes in Physics
, 5th ed. (
Springer International Publishing
,
Switzerland
,
2017
), pp.
137
and 158-160.
20.
We used the mass and the dimensions given from the technical data sheet by Apple Inc., USA, https://www.apple.com/de/iphone/compare/.
21.
W.
Schenk
and
F.
Krämer
,
Physikalisches Grundpraktikum
, 14th ed. (
Springer Spektrum
,
Wiesbaden
,
2014
), p.
378
.
22.
N. J.
Simon
,
S.
Drexler
, and
R. P.
Reed
, “Properties of Copper and Copper Alloys at Cryogenic Temperatures,”
U.S. Department of Commerce Technology Administration
(
National Institute of Standards and Technology
, February
1992
).

Andreas Kapsstudied 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.