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A medieval clock made out of simple materials
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2008 Eur. J. Phys. 29 799
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IOP PUBLISHING
EUROPEAN JOURNAL OF PHYSICS
Eur. J. Phys. 29 (2008) 799–814
doi:10.1088/0143-0807/29/4/013
A medieval clock made out of simple
materials
B Danese and S Oss
Dipartimento di Fisica, Università di Trento, Via Sommarive 14, 38100 Trento, Italy
E-mail: [email protected] and [email protected]
Received 4 April 2008, in final form 5 April 2008
Published 16 June 2008
Online at stacks.iop.org/EJP/29/799
Abstract
A cheap replica of the verge-and-foliot clock has been built from simple
materials. It is a didactic tool of great power for physics teaching at every
stage of schooling, in particular at university level. An account is given of its
construction and its working principles, together with motivated examples of a
few activities.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Have you ever seen a medieval clock in action? It will look incredible to whom has
never seen it, but from the rhythmic and continuous vibration of the daemon of the
balance beam, from the wriggling out of the escapement while every other part seems
immobile, it springs such a sensation of living thing that one is enchanted. It is a
show one never grows tired to assist to: infinitely more suggestive than a swinging
pendulum or than a rapidly beating modern clock. The world of humanities was in
ecstasy and struck with marvel.
So wrote Antonio Simoni in his massive work on the history of horology [1]. Not only
the world of humanities was struck with marvel, as is evident from many accounts [2], but also
the world of technology from which it stemmed. The mechanical clock has been regarded as
a ‘turning point of western technology’ [3, 4] and the simple model here proposed, in fact,
was developed in a course for middle school technology teachers. But most importantly, the
weight-driven medieval clock with the verge-and-foliot escapement is a fascinating object for
the teaching of physics at various levels, from middle school to university.
While the potential of the pendulum clock for science and physics education has been
fully developed [5, 6], the educational value of the verge-and-foliot clock has been accounted
for mostly in the field of control theory [7–9] and history of science [10–12]. The main object
c 2008 IOP Publishing Ltd Printed in the UK
0143-0807/08/040799+16$30.00 799
800
B Danese and S Oss
Figure 1. The driving-weight clock out of simple materials.
and novelty of this paper is to provide relevant examples of the use of the verge-and-foliot
clock as a tool for physics education.
For instance, middle schools are the places where the clock can be built hands-on from
simple materials and used for scientific explorations [13]. At middle school, clocks of many
types [14] can be described by means of drawings and causal explanations, and can be used in
discussing energy and machines. At high school every topic of introductory mechanics—from
levers to circular motions, from collisions to potential energy—finds an illustration in the
clock. It allows many activities of data taking and data analysis to be carried out using simple
chronometers and rulers. It is also a most interesting topic for the history of science and for a
visit to a museum [15]. It is at the university level that a deepened didactic use of the clock may
take place. There the clock can be completely described by means of analytical mechanics
with the desired degree of approximation. More detailed experiments can be carried out by
means of electronic sensors and automatic data acquisition, as is shown in this paper. In fact,
the clock itself is an oscillator, useful to illustrate topics like feedback and limit cycles [7–9].
In this paper we describe the construction and the working principles of the clock, together
with motivated examples of a few activities to be carried out at various stages of schooling.
The focal point of our perspective is university-level physics, thus the pre-university activities
we present should be considered in this perspective.
2. Building and principle of operation
The verge-and-foliot escapement was an invention of radical originality. The
complexity of its functioning and the ingenuity with which it has been devised far
surpassed all previous mechanical inventions. Its characteristic feature, the carefully
tuned dynamic interaction of several matching parts, is almost impossible to describe
in words or to illustrate in a two-dimensional picture; to fully comprehend it, one
should handle a working specimen [10].
A medieval clock made out of simple materials
801
Hence, the need for a verge-and-foliot clock built from simple materials. Our working
model is shown in figure 1. It is built out of toothpicks, cardboard, a straw, a popsicle stick,
sewing thread and a couple of screws. The frame is made with MeccanoTM pieces, but a pasta
box or a shoe box would work as well. The operation of the mechanism is the following.
(a) The driving weight (here a film can with a few coins inside, but any small object of
appropriate mass may work as well) is suspended by sewing thread coiled around the
shaft (a wooden skewer). At one end of the shaft, the crown or geared wheel (cardboard
disc) with its pegs (toothpicks) is pulled in rotation by the fall of the driving weight. A
peg of the crown strikes the upper palette (a popsicle stick piece) of the verge (a straw)
and the motion of the wheel is checked by the inertia of the verge, the foliot (a wooden
skewer) and the balance weights (pieces of key-holders).
(b) The driving weight slowly turns the verge and foliot until the peg has pushed the palette
out of the way (escapement) and the driving weight can, for a while, fall freely.
(c) The turning of the verge has, however, brought the lower palette between the pegs of the
wheel so that the next lower peg, moving in the opposite direction to those on top, almost
immediately strikes the lower palette. The turning motion of foliot and balance weights
and the fall of the driving weight are thus slowed down.
(d) Once more a peg pushes the palette and the whole verge and foliot, but this time in the
opposite direction so that the upper palette is brought back again into mesh with the pegs
of the wheel.
The process then repeats itself until the driving weight reaches the ground, with the foliot
and balance weights swinging to and fro.
2.1. The making of the clock or science and technology in middle school
Building such a device will give great satisfaction and many things to learn to students from
middle school to university (and to their teachers too). The brilliancy of the mechanical
insight that the motions in opposite directions of the upper and lower parts of the crown can be
converted into the cyclic to-and-fro motion of the verge and foliot will guide their reasoning
on quantitative aspects of the phenomenon (see figure 2).
How many pegs are required in the crown, for instance? What about the angle between
the palettes? What about peg length and crown–verge distance? Different inquiry paths are
open: geometrical reasoning, building of a test model, replication of the clock built by the
teacher, including its variations.
The technique of orthogonal projections—very common in Italian middle schools—can
be highly useful in this 3D reasoning. One discovers that if the number of pegs is even, the
upper peg escaping from the upper palette would have a symmetrical lower peg pushing the
lower palette in the opposite direction. This would block the clock, and one should choose an
odd number of pegs, for example, five. The principle of operation is very well shown with
five pegs and the construction of a pentagon, moreover, is a very beautiful ‘ruler and compass’
exercise.
The whole cause-and-effect analysis of the mechanism is stimulating for the student,
as well as the recognition of dynamical principles embodied in the construction. The most
striking one is action–reaction: when crown pegs push the palettes and turn the verge, then
palettes push the crown pegs, and the system shaft + crown + pegs slowly slides away from
the verge. Therefore, a block to avoid the sliding of the shaft along its rotation axis is needed
in the construction.
The clock builder also discovers the importance of the bond between crown and shaft,
and will try to fix it with plastic tape, glue or plasticine. The student may also find profit in the
802
B Danese and S Oss
Figure 2. The kinematical insight of the verge-and-foliot clock: the motions in opposite directions
of the upper and lower parts of the crown can be converted into the cyclic to-and-fro motion of the
verge and foliot.
device’s characterization with his own watch or chronometer and check the repeatability of its
tic-tacks. This activity may be considered not only as a control of the performance, but also as
the calibration of an instrument, since different clocks may be arranged to beat the same time.
2.2. Simple models of the clock or high-school physics
In high-school physics the mechanical clock can be used at least in three important ways: (a)
as an illustration or starting point for many topics connected with motion, (b) as a source for
several ‘Kapitza problems’ [16] and (c) as a landmark for narrating the history of physics.
The topics for which the clock provides illustrations or starting points comprise the
following:
• Uniform, accelerated, circular and periodic motion. The motion of the driving weight, in
fact, keeps accelerating and stopping and time is divided in small equal units, which are
eventually counted. However, the motion of the falling weight in a given time unit is not
uniform. Crown pegs may also illustrate circular and harmonic motion.
• Focusing on balance weights; references can be made to levers, to pendulums of different
lengths, to angular momentum and torques. There is the typical example of opening the
door far from or close to the door hinge. Other comparisons come from simple pendulums
and metronomes.
• Forces. Both external forces, like gravity pulling down the weight, and internal forces,
like the tension in the string or the forces between peg and palette obviously have an
essential role in the understanding of the clock.
• Collisions. The peg and palette hit each other, and their collision may be investigated
changing the quantity of motion of the respective systems by means of the attached weight.
One may therefore have a palette bouncing back or a palette pushing peg and wheel in
the opposite direction.
A medieval clock made out of simple materials
803
Figure 3. Sketches for problems 1 and 2.
The verge-and-foliot clock is an inexhaustible source of exercises, of ‘Kapitza Problems’.
The great scientist and teacher stressed the importance of this type of problem at the first
International Congress on the Education of Teachers of Physics in Secondary Schools [17].
It is very important to have laboratories and seminars and to solve problems which
will encourage independent thinking. The exercises given to schoolboys are not
always good for this. Mostly such problems give the pupil the data and he is to put
the information into the proper formula to produce the correct answer. The work is
to find the correct formula. This is not really independent thinking.
I used to give my students at the university a different kind of problem. I will simplify
a few examples for secondary-school levels. (1) The power of a motor necessary to
move a pump will give a jet of water large enough to extinguish a fire in a six-story
building. With this problem the pupil himself must judge the height of the building,
and where to put the pump. Each pupil may decide differently on these points, and it
is easy to see which is better (2). What is the size of a convex lens which will focus
sunlight in a spot where you will make a furnace. The pupil must find out what heat
is necessary and how good the focus must be in determining the size of the lens, and
once again he will be thinking out the whole problem [18].
The verge-and-foliot clock may be profitably used as an object around which many typical
exercises become more meaningful and may serve as steps for the construction of a ‘working
theory of the clock’. Such is the kind of independent thinking, or ‘thinking out the whole
problem’, that is the objective of the above-mentioned Kapitza problems.
Typical exercises in this context are those where a little mass m suspended to a string
pulls attached masses or systems bigger than hers, with negligible friction. Let us single out
a couple of examples:
(1) the attached system is a big mass M;
(2) the attached system is a verge, of radius r, and foliot, of length l and two masses M (see
figure 3).
In the first case one can work out Newton’s II law ṗ = F (that is ma = F ) for both
masses. Let us underline the conceptual importance of grouping variation of momentum (p
for mass m and P for mass M) on one side of the equations and forces on the other,
ṗ = mg − T
ma = mg − T
which is
(1)
Ma = T ,
Ṗ = T
arriving at
m
g.
(2)
a=
m+M
804
B Danese and S Oss
Table 1. Data for foliot lengths and time periods of complete rounds of the crown (10 tic-tacks).
Partial times of 10 tic-tacks (s)
Length l (cm)
10
20
30
40
Time (mean) t (s)
18.4 ± 0.1
12.7 ± 0.1
5.6 ± 0.1
13.71
9.68
3.88
27.44
19.25
7.93
41.08
28.66
11.70
54.92
38.06
15.46
13.7 ± 0.1
9.5 ± 0.1
3.9 ± 0.1
In the classroom the approximations on the friction of M (negligible), on the string
(inextensible) and on the pulley (weightless) can be discussed after obtaining this result, so
that their bearing on the result may be readily shown. Friction, for example, may be added in
the equation and numerical tests for the justification of the approximation can be performed.
From the expression for the acceleration one may proceed to consider that the time
necessary for M to travel a fixed distance d is given by
2d M + m
t=
.
(3)
g
m
In the second case, to be coupled with ṗ = F , one has the equation for the variation of
angular momentum of the verge and foliot:
ṗ = mg − T
ma = mg − T
ma = mg − T
and
then
(4)
which is
1
1
2
Ml
α
=
T
r
Ml 2 a = T r 2 .
L̇ = τ
2
2
The accelerations are linked together by the consideration that α · r = a. Again one has
to solve a system of two equations with two unknowns, a and T. One obtains
a=
m
m+
l2
M
2 r2
g
that may be approximated with
a≈
2mr 2
g
Ml 2
(5)
or
2mr
g.
(6)
Ml 2
From the expression for the angular acceleration (6) one may obtain the time necessary
for the foliot to cover a given angle θ :
θ M
t ≈l
.
(7)
gr m
∝≈
The time interval necessary for turning the verge and foliot a given angle, while in square
root relation with the mass ratio as before, is in linear relation with the foliot length. The
relation may be confronted with experiment. Measuring different times for varying lengths
of the foliot is very good for the topic of measurement, imprecision bars and basic statistical
analysis. It can also be very good for checking the performance of the clock.
We present a set of simple data taken for different lengths of the foliot, by means of the
chronometer of a mobile phone which allowed us to measure up to four partial times.
Three different foliot lengths, the corresponding raw data of the partial times of one
complete round of the wheel (10 tic-tacks) along with the respective mean values, are shown
in table 1. The data are plotted in figure 4. For clarity the uncertainty bars on the foliot lengths
have been propagated on the time axis. The relation between times and lengths is linear:
t = 0.73l.
(8)
A medieval clock made out of simple materials
805
Figure 4. Plot of foliot lengths versus time intervals of complete rounds of the crown (10 tic-tacks).
Students are struck by the fact that a real clock can be built from scratch, and that its
regular behaviour (the linear relation between the time intervals and foliot lengths) can really
be accounted for by means of formulae.
The ‘rough prediction’ plotted in figure 4 is estimated by means of equation (7) with the
substitution of r with an effective radius r∗ from the interplay of the various gears:
r∗ =
r 0 l0
rd
(9)
where r0 is the radius of the shaft, rd is the radius of the crown wheel and l0 is the distance
between the verge axis and the end of the peg (see figure 5).
From the values and uncertainty bars of the various parameters θ , m, M, r∗ we obtain
for the constant in (8) the value of 0.50 s cm−1 (a range 0.41–0.59 s cm−1). The rather
gross value may be refined taking into account the time needed to slow down the foliot at
the impact and the fact that acceleration is not constant, as is discussed later. We thus obtain
0.78 s cm−1 for the constant in equation (8), and a predicted range of 0.70–0.86 s cm−1, in
good agreement with the experimental value of 0.73 s cm−1.
In this way, the construction and solution of several Kapitza problems lead to the
comprehension of various aspects of the working of the clock. Further problems, for example,
may contain friction or gears and pulleys with different radii.
The point of view of mechanical work is also a very interesting one. While the suspended
weight falls, the shaft and the crown are slowly set in motion, and so are the verge and the
foliot with its balance weights. The energy of motion of the latter system is then converted
in halting with a collision of the crown and the falling weight. It is for this reason that the
word feedback has been adopted in the description of this ancient clock, which has thus been
called a double feedback clock. Thus, it happens that a key word of electronics characterizes
a device whose origin is shrouded in mystery.
Once the clock has been explored quantitatively and modelled, its use in connection with
the history of science should seem very interesting at high-school level. The history of devices
806
B Danese and S Oss
Figure 5. Sketch of the clock (without foliot). The upper palette has just escaped at the angle θ = θ E.
for the measurement of time was valued, for instance, by Thomas Young, who in his Lectures
on Natural Philosophy and the Mechanical Arts [19] compiled a whole history of timekeeping.
Of the medieval clock, Young gave the following description:
The alternate motion of a balance, thrown backwards and forwards by the successive
actions of a wheel impelling its pallets, is also capable of producing a degree of
uniformity in the motion of the wheel; for the force operating on the pallet is
consumed in destroying a velocity in one direction and in generating a velocity
in the contrary direction; and the space in which it acts being nearly the same in all
cases, the velocity generated will also be nearly the same at all times, as long as the
force remains the same [19].
This beautiful mechanism had—according to Simoni [1]—been developed by a
blacksmith or astrolabe-maker (the technology necessary for clock building was the one
for astrolabes building, mastered at the time) or monk (the first widespread use of mechanical
clocks was in monasteries), probably from armillary spheres and machines representing the
motion of the heavenly spheres. This truly creative invention shows once more how in a true
cultural outlook mechanical objects and the universal laws of mechanics are connected.
3. Clock description by means of classical mechanics
At university level the quantitative description of the whole clock can be undertaken as a
problem of analytical mechanics. The clock may be seen as composed by three moving bodies
connected by two sets of internal forces, such three parts being
• the falling weight, of mass m, moving downwards with acceleration a;
• the rotating crown + pegs + shaft, of inertia moment Id, covering the angle ϕ with angular
velocity ωd;
• the oscillating verge + palettes + foliot + weights, of inertia moment I0, covering the angle
θ with angular velocity ω0.
The internal forces are
• T, the string tension which pulls up the weight m and torques the shaft (whose radius is
r0);
• F, the force between the end of a peg and a palette.
A medieval clock made out of simple materials
807
Table 2. Data for foliot lengths, inertia moments, angular velocities and angular moments.
Length
l (cm)
I0 (calculated)
(g cm2)
ωS (calculated)
(rad s−1)
L0 (calculated)
(g cm2 rad s−1)
Ld (calculated)
(g cm2 rad s−1)
18.4
12.7
5.6
1693 ± 18
806 ± 13
157 ± 6
1.65 ± 0.3
2.4 ± 0.5
5.4 ± 1
2794 ± 533
1929 ± 368
850 ± 162
122 ± 28
177 ± 40
402 ± 91
3.1. Inertia moment
All the simplest moments of inertia are strikingly embodied in the verge-and-foliot clock,
and the latter may serve as a beautiful introduction to the former. Moreover, the estimation
of inertia moments involved in the clock is important for its description. One obtains Id =
175 g cm2, I0 = 1693 g cm2 (at maximum foliot length, 806 and 157 g cm2 at inferior lengths,
see table 2) where the momenta of inertia of a horizontal rod (foliot and palettes), a cylinder
(shaft, crown and pegs), a hollow cylinder (verge) as well as the parallel axis theorem (for
pegs and palettes) are employed.
3.2. Angle of escapement
In the tic-tacking of the clock we distinguish three phases:
• escapement, when the peg initially at rest pushes the upper palette from θ = 0 to θ = θ E,
(in which θ E denotes the maximum angle before the peg leaves the upper palette, i.e. the
escapement itself);
• free fly, when the foliot, once ‘escaped’, turns at constant speed and the crown freely
accelerates until the lower palette impacts on the lower peg; we consider this phase an
instantaneous one;
• collision, the instantaneous impact in which kinetic energy of foliot and wheel are
exchanged, and the following slowing down of wheel and foliot.
We begin with the analysis of the escapement phase and thus proceed to write a system of
three equations in three unknowns, which is necessary for the description of a system of three
moving bodies:
⎧
⎨ma = mg − T
Id αd = T r0 − rd F cos θ cos ϕ
(10)
⎩
I0 α0 = F l0 / cos θ.
Furthermore, the system is a connected mechanism and two relations between a, α d and
α 0 hold:
a
(between a and αd );
(11)
αd =
r0
and the relation between α d and α 0, which is more cumbersome:
or
αd rd2 − l02 tan2 θ = α0 l0 .
(12)
αd rd cos ϕ = α0 l0
From the geometrical relation between ϕ and θ ,
rd sin ϕ = l0 tan θ,
one also obtains an expression for the foliot ‘arm’ l0 and the angle of escapement θ E:
π
l0 = rd sin cotan θE .
5
(13)
(14)
808
B Danese and S Oss
Thus, the wheel and shaft can be carefully placed so that l0 matches the actual angle
between the palettes. In our case θ E ≈ 70◦ . However, our pegs are not very precise and the
escapement takes place at angles between 70◦ and 50◦ . We thus set l0 = 1.7 cm.
With the relations (11) and (12) one has now obtained the desired system of three
equations—corresponding to three objects—and three unknowns—corresponding to one
degree of freedom, a (or α d or α 0), and the two internal forces T and F:
⎧
ma + T = mg
⎪
⎪
⎪
⎪
⎨ Id
a − r0 T + rd F cos θ cos ϕ = 0
(15)
r0
⎪
2
⎪
l
r
cos
ϕ
I
⎪
⎪
⎩ 0 d
a − 0 F = 0.
r0
cos θ
3.3. The differential equation of the first escapement
Once the solution of (15) is found, from a one obtains
r02 m
r 0 l0
θ̈ = g
r 2 cos2 ϕ cos2 θ
rd cos ϕ
Id + r02 m + I0 d l 2
(16)
0
which can be greatly simplified if these approximations hold:
rd2 cos2 ϕ cos2 θ
l02
2
r cos2 ϕ cos2 θ
r02 m I0 d
l02
cos ϕ ≈ 1.
Id I0
(17)
(18)
(19)
In fact, even in the worst case (the greatest values for ϕ and θ , ϕ E = 36◦ and θ E = 70◦ ),
one has
mr02 ≈ 1.5,
rd2
cos2 ϕE cos2 θE ≈ 5.5,
l02
cos ϕE ≈ 0.81.
(20)
With the above-mentioned approximations, one obtains
θ̈ =
mgr0 l0
.
I0 rd cos2 θ
(21)
The various parameters can be summarized in the constant A of dimensions T−2:
A=
gmr0 l0
.
I0 rd
(22)
In our case, A = 1.5 ± 0.6 s−2, see table 3.
The description of the escapement phase is therefore reduced to the Cauchy problem
θ̈ =
A
cos2 θ
(23)
with initial conditions θ̇ (0) = 0 and θ (0) = 0.
Before undertaking its solution, let us ponder the plot of the function cos−2θ , which thus
also represents θ̈ in units of A (figure 6). Thus, the whole problem may be approximated with
θ̈ = A
(24)
A medieval clock made out of simple materials
809
Figure 6. Plot of y = cos−2 θ . It is 2.5 at 50◦ and almost 10 at 70◦ .
Table 3. Data for foliot lengths and angular accelerations, and times of one-round periods.
Length
l (cm)
A
(calculated)
(rad s−2)
10tS
(calculated)
(s)
AN
(calculated)
(rad s−2)
10tD
(calculated)
(s)
10tS + 10tD
(calculated)
(s)
10t
(measured)
(s)
18.4
12.7
5.6
1.5 ± 0.3
3.2 ± 0.6
16.5 ± 3
10.8 ± 2.2
7.4 ± 1.5
3.3 ± 0.7
2.4 ± 0.5
5±1
26 ± 5
6.8 ± 1.4
4.7 ± 0.9
2.0 ± 0.4
17.6 ± 3.6
12.1 ± 2.4
5.3 ± 1.1
13.7 ± 0.1
9.5 ± 0.1
3.9 ± 0.1
for small angles θ , that is, for a high number of pegs. While typical medieval clocks have
crowns with a high number of pegs, this is not our case. We must deal with equation (23).
The first integration is easily achieved, and one obtains the angular velocity
√ √
ω0 = 2A tan θ ,
(25)
that is, the Cauchy problem
√ √
θ̇ = 2A tan θ
with the remaining initial condition θ (0) = 0.
The integration of the latter is trickier but nonetheless exact and is
√
√
−2 arctan(1 −n 2 tan θ ) + 2 arctan(1 + 2 tan θ )
tan θ + √2 tan θ + 1 √
+ ln √
=4 At
− tan θ + 2 tan θ − 1 (26)
(27)
which we write as
√
F (θ ) = 4 At.
(28)
We thus arrive at the desired expression t = t(θ ), from which we can obtain θ = θ (t).
Before studying the latter, let us consider once more the approximation for small angles θ̈ = A.
It would follow
A
θ̇ = At
and
θ = t 2.
(29)
2
810
B Danese and S Oss
Figure 7. Plot of θ = θ (t) of the first escapement from equation (28) (continuous line) and from
equation (30) (dotted line).
The latter could be rewritten as
√
√
4 2θ = 4 At
(30)
which compares very well with (28). The two functions (28) and (30) are plotted in figure 7
as θ = θ (t).
It turns out, therefore, that even if the acceleration at wide angles is up to ten times its
initial value, the discrepancy between the time of escapement in the two models is of the order
of only 6%. The approximated one may thus be safely used at high school level, since it yields
compatible results.
3.4. Measurement of the first escapement with electronic acquisition
Electronic acquisition allows us to measure the rotation of the foliot and verify equation (28)
plotted in figure 7. We use the Pasco Rotary Motion Sensor RMS-BTD mechanically coupled
with the verge . A new set of data is acquired in the new configuration. Oscillations are slower
(the fit gives t = 0.95l) by a factor of 1.29 (see figure 8).
The two estimations in figure 7 need therefore to be rescaled by the same factor, 1.29.
For examples, the foliot should now arrive at the angle θ = 60◦ in 1.45 or 1.51 s, and not as
before in 1.12 or 1.17 s.
We acquired the data of the first escapement with a resolution of 1440 divisions/rotation
and a sampling rate of 200 Hz. The data are plotted in figure 9. The agreement between data
and theory is remarkable. We see that the escapement takes place at 58◦ and after the sudden
impact the motion of the foliot is slowed down with almost uniform acceleration.
An acquisition for a more prolonged time is plotted in figure 10. Single tic-tacks are
visible: the peaks are when the foliot is stopped by the lower palette and the bottoms are when
it is stopped by the upper palette. The different heights of these peaks are suggestive of the
misalignments among the five pegs. However, the cycle of ten tic-tacks (five peaks and five
bottoms) is clearly visible.
A medieval clock made out of simple materials
811
Figure 8. Plot of time intervals of complete rounds of the crown versus foliot lengths (10 tic-tacks).
Figure 9. Plot of θ = θ (t) of the first escapement from equation (28) (continuous line) and from
equation (30) (dotted line).
Other information one can gather from these measurements is that the first escapement is
somewhat slower than the successive ones, when the clock reaches its limit cycle.
3.5. The time interval of the first escapement
From inversion of (28) we obtain θ = θ (t), which is plotted in figures 7 and 9 together with
the approximated function from (30). For θ = θ S one obtains the time of escapement tS,
tS =
F (θS )
√ ,
4 A
(31)
812
B Danese and S Oss
Figure 10. Foliot angle acquired on-line with the Pasco rotary motion sensor.
or, within 6% accuracy,
tS ≈ 2θS /A,
(32)
which is also consistent with equation (7), obtained in the simplified high-school Kapitza
problem.
3.6. The collision
The correction to the time interval used in figure 3 takes into account the time in which the
driving weight and the crown stop the motion of the foliot after the escapement and the impact.
Thus, one has to take into account the sudden drop due to the impact and the deceleration from
this residual velocity to 0. As has been done so far, this question can be addressed by means
of Kapitsa problems or by a detailed analysis that can also be tested with data acquisition.
We confine ourselves to a few considerations, referred to the set of data in table 1, figure 4,
discussed in section 2.2. In the case of the longest foliot, after the escapement the verge+foliot
is turning with angular velocity ωS ≈ 95◦ s−1 (a value consistent with the data in figure 9). Its
angular moment, therefore, is
L0 ≈ I0 ωS ≈ 2794 g cm2 rad s−1 .
(33)
The crown, once the peg has escaped, has an angular momentum
Ld ≈ Id ωd ≈ 122 g cm2 rad s−1 .
(34)
In the simplest approximation we thus regard as unchanged the angular velocity of the
foliot just after the impact with the peg. The correction to the time interval is therefore mainly
due to the deceleration, when the foliot pushes the crown backwards. In this case the efficient
radius is not like equation (9): instead of the partial arm l0 (in our case 1.7 cm), where the peg
pushes the palette, one has to take into account l1 (in our case 2.7 cm), the whole length of the
palette that is pushing the peg:
r 0 l1
.
(35)
r∗ =
rd
A medieval clock made out of simple materials
813
The calculated times are shown in table 3. The estimation of one-round period sums up
ten time intervals due to escapement tS and ten time intervals due to deceleration tD. The value
is slightly higher than that actually measured, the real clock goes quicker than the modelled
one. This could be taken into account using the exact F(θ ) from (27), which reduces tS by
5–6%, as is done in figure 4. The discrepancy may be further reduced taking into account the
fact that ω of the foliot is diminished by the impact with the peg, which reduces tD by 6–60%
according to the length of the foliot.
4. Conclusion
We have shown the didactic utility of a mechanical clock made out of simple materials. It
may be used in different ways according to the level of school, but in particular at university.
It can be used in countless ways and for countless exercises.
We had fun in letting the suspended weight travel down for three floors. However, further
improvements can still be made to our simple mechanism. First of all, the addition of more
wheels and a hand. The wheel with pegs could be re-shaped and imitate the original medieval
crown, with its characteristic teeth, which may be considered as the link between gears made
of pegs and modern gears.
Of course, the teacher may also elaborate on the metaphor of the world as a clock, and
its many examples in the literature. However, from the cultural point of view, we are content
to underline that the clock allows us to use the word mechanics in both its meanings, that of
mathematical physics and that of gears and shafts, together.
Acknowledgments
We would like to thank the group of technology teachers for their participation in the activity
and useful observations.
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