  The Wilberforce pendulum is a great experience for illustrating important
properties of coupled oscillatory systems, such as normal modes and beat
phenomena, in physics courses. A helical spring attached to a mass comprises
this simple but colorful device that features longitudinal and torsional
oscillations, catching the eye of students and teachers, and offering a
fantastic environment to study coupled oscillations at qualitative and
quantitative levels. However, the experimental setups that are commonly used to
simultaneously acquire both oscillation types can be cumbersome and/or costly,
often requiring two synchronised cameras, or motion sensors and photo-gates
integrated by an expensive analog-to-digital converter. Here, we show that a
smartphone can be the perfect device for recording the oscillations of a
Wilberforce pendulum composed of just an educational helical spring and a can.
Our setup employs a smartphone's accelerator and gyroscope to acquire data from
the longitudinal and torsional oscillations, respectively. This setup can
successfully record both normal modes and beats. Furthermore, we show that our
experimentally obtained time-series and power spectra are in great agreement
with simulations of the motion equations. In addition, we critically analyse
how to set the initial conditions to observe both normal modes and beats. We
believe our study contributes to bring the striking physics of the Wilberforce
pendulum closer to undergraduate classrooms with low-cost, highly-accessible
materials.
