The Wilberforce pendulum:
a complete analysis through RTL
and modelling
The Wilberforce [1] pendulum is a didactical device often used in class demonstration to show the
amazing phenomenon of coupled oscillations (rotational and longitudinal) producing beats in a special
mass-spring set up. The phenomenon is particularly surprising for a distant observer who easily detects
the longitudinal motion but may skip the presence of rotation. To him (her) the vertical oscillation
appear to damp out completely and then it rises again without external action (as if an invisible force
would come in). In this contribution we propose a revised version of the classical experiment, as an
example of an efficient didactical use of modern technologies, suggesting a deeper understanding of
phenomena through real time data acquisition and advanced computer modelling.
Our experimental set up
This device picked up the curiosity of many [2] , among which Arnold
Sommerfeld, who gave us a complete theoretical treatment [3] .
Recently it was studied also with the aid of an RTL system (sonar
interfaced to a PC) by measuring the vertical oscillation [4]. Nobody,
however, completely characterized the Wilberforce pendulum motion
with RTL (both longitudinal and rotational coupled oscillations). In this
contribution we show how it may be done by using, besides the sonar,
also a Non-Contact Rotation Sensor [5] (NCRS).
Our NCRS is an optical probe that exploits the intensity modulation of a
polarized light beam (emitted by a LED placed behind a rotating
Polaroid driven by a motor). The light beam, after a reflection by a
second Polaroid attached to the rotating target, is detected by a
photodiode. A lock-in technique and some more electronics gives an
output voltage that is linearly proportional to the rotation angle.
The pendulum is produced by LABTREK [6] with an helicoidal spring
and a cylindrical brass mass of about 0.5 kg. The upper end of the spring
is blocked to a rigid holder, and the cylinder is blocked to the lower end
of the spring. A reflector screwed to the lower cylinder end is covered
by a Polaroid disc.
Both the sonar and the NCRS are placed at ground (side by side) so that
both the ultrasonic beam and the IR light beam reach the target reflector
(the useful distance range is between 0.5 m and 1.0 m).
Experimental results
An example of the recorded rotation angle and of the vertical
displacement obtained using a Personal Computer and the LabPro
Fig. 1: The pendulum with interface driven by LoggerPro [7]:
Fig. 2: Displacement and angle as functions of time
The beats in figure 2 are evident: the vertical oscillation slowly dumps out while rotational oscillation
rises, then the process reverses (approximately every 1/2 minute) with the energy associated to each
oscillation that transfers from one motion to the other, with negligible dissipation. The two oscillations
must have the same period (in our case about 0.91 s) in order to achieve a complete energy exchange.
By plotting the angle as a function of displacement we see that the phase angle between the two signals
covers a whole region in the phase space.
The total energy in each motion is made of an elastic potential term Ee and of a kinetic term [8] Ek.
For the longitudinal motion: ET= EkT + EeT= (m/2)v2+(k/2)z2 (where m is the mass, v the velocity, k
the elastic constant, and z the displacement from the equilibrium position).
For the rotational motion: ER= EkR + EeR= (I/2)ω2+(D/2)α2 (where I is the inertial momentum,
ω=dα/dt the angular velocity, D the torsional constant and α the rotation angle).
The “total pendulum energy” is therefore E= ET + ER =(m/2)v2+(k/2)z2 +(I/2)ω2+(D/2)α2 .
In order to evaluate the behaviour of the various terms one must know the values of the four parameters
m,k,I,D. To measure the mass m we only need a scale, or a calibrated force sensor. To calculate k we
may measure the spring elongation under a known force (e.g. the cylinder weight mg). We may also
calculate the inertial momentum I for a simple geometry [9]: for a cylinder of radius R and mass m it is
I=mR2/2 .The torsion constant D may be calculated from the measured period T =2π/ω and the relation
that gives the angular velocity: ω2=D/I, or D=I(2π/T)2 .
Fig. 3: Elastic and kinetic energies for longitudinal (EeT, EcT) and rotational (EeR, EcR) motions
The time evolution of elastic and kinetic energies may be calculated from the measured angles and
displacements, as shown in figure 3 .
By adding elastic and kinetic terms we obtain the “total energy for each motion” (figure 4).
Figure 4: Translational (ET), rotational (ER) energies and their sum as functions of time
By finally adding the two “pulsed” terms ET(t) and ER(t) we obtain the total energy that appears to
decrease very slowly in time (in figure 4 we see that the time constant of the exponential decay is about
170 s). The oscillations that modulate the exponential decay, at this stage of our investigation, may be
attributed to “noise” or to approximation in the parameters’ values used to calculate the total energy.
We will see later that there is a different explanation …
Fig. 5: Fourier Transform plot (two frequencies appear)
Even if at first sight the two oscillations appear to have exactly the same frequency, a Fourier analysis
(either of the longitudinal or of the rotational motion) shows that two frequencies are present (figure 5):
this is due to the coupling between two normal modes that can be excited by properly choosing the
initial conditions, as can be better shown by using a simulation.
We may build a simulation of the whole process by using recursive numerical computation, either
exploiting the graphic calculator tools for differential calculus, or on a PC using software applications
like Stella (or MathLab or Mathematica).
We write a set of differential equations for the coupled oscillators, accounting for a finite coupling
constant [10]ε between rotation and displacement [11].
For longitudinal motion the main restoring force is –kz, but accounting for a coupling with rotational
motion we add the term –εα : the longitudinal acceleration is therefore d2z/dt2= –(kz+εα)/m .
For rotational motion the main restoring torque is –Dα, but accounting for a coupling with longitudinal
motion we must add a term –εz : the angular acceleration is therefore d2α/dt2 = –(Dα+εz)/I .
Fig. 6: Simulation obtained with a TI graphic calculator
This pair of equations may be solved, as shown in figure 6, using a TI-Voyage200 graphic calculator
(with the parameters derived from experimental data and a properly chosen value for the coupling
constant ε). The value of the coupling constant may also be evaluated through a detailed analysis of
the system and by knowing details [12] of the spring (where the coupling takes place), but the
simulation brings-in a great help: the ε value determines the beats period. The stronger is the coupling
the faster becomes the energy transfer across modes, the shorter becomes the beating period.
Working with graphic calculators may be amusing and appealing (due to their great portability) but for
such complex computations they may become extremely slow…
For didactical purpose it is better to switch to a PC-based application: we choose Stella (which joins
power to easiness of use) to work out the simulation shown in figure 9 for displacement, angle, phase
and energies respectively.
Fig. 7: Simulated displacement and angle vs. time (z0=5 cm and α0=0 rad)
Figure 7 shows well the beating effect as well as the behaviour of the phase evolution. The last plot
shows how the energy is exchanged between rotational and longitudinal oscillations, while total energy
slowly decays due to dissipation.
One advantage of Stella is the easiness in setting the initial conditions, which are quite important in
coupled oscillators. All the above results are in fact related to special initial conditions (zero for all
velocities and for initial angle, and finite initial displacement). This is what usually happens when we
start playing with the Wilberforce pendulum.
However if we choose different values (Fig. 8) we soon realize that the pendulum behaviour does
sensibly change: for particular values of initial angle and displacement we may completely separate the
two modes (in the sense that the energy stays substantially constant for each mode, and beating
Here we distinguish between oscillation type and fundamental mode. A fundamental mode is made of
both of longitudinal and of rotational oscillations: there are two modes, one where angle and
displacement are in phase and the other where they are in phase opposition. The two frequencies we
detected experimentally are related to these two modes, which can be excited separately by a proper
choice of initial conditions.
Fig. 8: Simulated behaviour with z0=5 cm and α0=2.85 rad
Also dissipation is easily handled through Stella: both in figure 7 and 8 the simulations account for
viscous forces (both in rotational and in longitudinal motion). In figure 8 the initial conditions are set to
reach a complete mode decoupling (i.e. we excite a single mode), and the graph of the energies
(rotational, translational and their sum) as a function of time shows that with these initial conditions no
energy is transferred from rotation to translation or viceversa, and both amplitudes decrease smoothly
in time. The phase plot shows that angle and displacement are now constantly in-phase. By reversing
the initial angle we obtain the same behaviour, but with a constant dephasing of π. These two initial
conditions are those that excite the two decoupled oscillator modes .
Fig. 9: Experimental behaviour with z0=2 cm and α0=1.14 rad
A record of the actual oscillations, taken with these initial conditions, is reported in figure 9, where the
FFT plot shows only one peak corresponding to the normal mode at higher frequency.
The slight oscillations that still appear in the simulated exponential decay of the total energy (in Figs. 7
and 8) tell us that something is missing in our model: because we introduced a coupling between modes
we must also account for the energy associated to the coupling terms!
This is simply done by adding to the elastic and kinetic terms the coupling energy Ec =εαz .
Fig 10 : Translational (1) , Rotational (2) and corrected Total (3) energy as functions of time
The results is shown in figure 10, where the oscillations (still visible in the energies associated to
rotation and to displacement) disappear in the total energy curve.
It is easy to show how the beating frequency (that equals the difference between the two peaks
observed in figure 5) depends on the value assigned to the coupling constant ε.
Also the simulation shows that, in presence of dissipation, a unique value of the ratio between the
angular and the longitudinal friction coefficients may produce total decoupling of the two modes: any
other value of this ratio will make impossible to produce pure oscillations without beatings!
In our experimental setup this matching was not perfect, leading to the slight residual mode coupling
shown in Fig. 9.
1. Named after R.L. Wilberforce, a physics demonstrator at the Cavendish Labs, who in 1894
showed how a cylindrical mass suspended to an helicoidal spring may undergo both rotational
and longitudinal oscillations, later publishing a paper on Philosophycal Magazine 38, 386
2. U. Köpf: Wilberforce’s pendulum revisited, Am.J.Phys. 58, 833 (1990), R.E. Berg, T.S.
Marshall: Wilberforce pendulum oscillations and normal modes, Am.J.Phys. 59, 32 (1991),
F.G.Karioris, Wilberforce pendulum, demonstration size: The Phys. Teacher 31 , 314 (1993).
3. A. Sommerfeld, Mechanics of deformable bodies: lectures on theoretical physics II, Academic,
New York, 1964.
4. E. Debowska, S. Jakubowicz, Z. Mazur: Computer visualization of the beating of a Wilberforce
pendulum Eur. J. Phys., 20, 89 (1999).
5. Made as prototype by Dusan Ponikvar (Ljubljana University), and summarized in Rev. Sci.
Instrum. 70, 2175 (1999).
6. https://www.labtrek.it/.
7. From Vernier Software http://www.vernier.com . Obviously similar interfaces and software,
like those from PASCO or CMA, may be used as well.
8. It may be shown that in this system the gravitational potential energy may be excluded from the
energy balance by an appropriate choice of the reference frame.
9. A more accurate measurement of the momentum of inertia Ix for a non-cylindrical mass may be
obtained through a double measurement of the period: with the mass by alone and then by
adding an annular cylinder of known mass and geometry (with well known value of inertial
momentum In). The two measured periods obey to the relations (T1/2π)2 = Ix/D ,( T2/2π)2 =
(Ix+In)/D giving: (T2/ T1)2=1+In/ Ix , and finally Ix= In T12 /( T22 – T12).
10. See ref 2 and 3.
11. For example http://www.phy.davidson.edu/StuHome/pecampbell/Wilberforce/Theory.htm.
12. E.g. we must know the pitch of the spring turns and the Young modulus of the spring material.