Pendulum
as a culturally rich topic
to teach physics
Igal Galili
Science Teaching Center
The Hebrew University of Jerusalem
Pendulum is a traditional item in physics curriculum
1. Pendulum is used to
practice measurements

2. Pendulum is used to test
students’ understanding of
force-motion relationship
Questions asked in matriculation examination:
Show/calculate the forces on the bob in…
Show/calculate the acceleration of the bob in…
Show/calculate the velocity of the bob in…
Calculate the period…
l
T  2π
g
Example:
Question from the matriculation examination of 1998
The focus is on the formalism and procedural knowledge
The required knowledge: Huygens’ formula
Example:
Question from the matriculation examination of 1991
The term (conical) pendulum is not even mentioned
Example: Question from the matriculation examination (Phys.lab.)
Galileo’s experiment with this setting is not mentioned
Example:
Question from the matriculation examination (Phys.lab.)
Students empirically investigate physical pendulum
without any theory and identification
Features of the current presentation
Used:
Missed:
 Pendulum presents the
case of vertical movement
on a circular path
 Mathematical Pendulum is
isochrornic (harmonic) only
approximately
 Mathematical (mandatory)
and physical (elected)
pendulums are addressed.
 Pendulum can measure time
 Pendulum can provide operational
definition of time
 Understanding of the
force-motion relationship
is tested
 Pendulum represents falling of
bodies (Galileo)
 “When period is asked
then the formula of
Huygens is applicable”
 Conical pendulum “explains” nonfalling of bodies - imitates satellite
(Hooke)
Pendulum
 What is it?
 What is it included for?
What does it represent?
Object…
System…
Model…
Theory …
The status of scientific knowledge
is ignored (hierarchy, importance…)
• A holistic view is neither provided nor required
What can be suggested?
Teaching physics as a discipline-culture
rather than a discipline.
Physics as a Discipline-Culture
body
nucleus
periphery
 Nucleus (center) – elements identifying the paradigms, concepts,
principles, axioms, rules of knowledge production
 Body area – elements of knowledge which are produced basing on
the rules of the nucleus
 Periphery (margins) – elements of knowledge which are at odds
with the nucleus
Holistic view on the subject of pendulum
within the discipline-culture framework:
1. What is the contribution of pendulum to
the principles of classical physics?
(nucleus)

2. What other conceptions of pendulum
are possible, were dismissed, and so on?
(periphery)
3. Usually standard problem solving is
trained and assessed
(body)
Periphery
Body
Nucleus
Examples of materials which could be introduced
Collision of two views on the nature of science
Nucleus
Galileo
Guidobaldo del Monte
1607-1545
Newton
Two conceptions:
Galileo’s versus Newton’s
Which one do we teach?
 Is Mathematical Pendulum isochrornic (harmonic) ?
... it must be remarked that one pendulum passes through its arcs of
180°, 160°, etc., in the same time that the other swings through its
10°, 8°, etc. …if two persons start to count the vibrations, the one
the large, the other the small, they will discover that after counting
tens and even hundreds they will not differ by a single vibration,
not even by a fraction of one.
Galileo, Dialogues Concerning Two New Sciences (1638)
 Pendulum can measure time (three meanings)
Timekeeping
(eventual)
Time-meaning
(operational)
Time-measuring
(instrumental)
Nucleus
• What makes pendulum moving?
Aristotle:
air
Periphery
Philoponus:
impetus
Galileo:
gravity as a quality of bodies
Descartes:
vortices of ether
Newton:
Gravitational force
Einstein:
Space Time curvature
• Pendulum entered science in the Medieval European
science of the 14th century
Jean Buridan
(1290-1360)
Albert of Saxony
(1316-1390)
Nicole Oresme
(1320-1382)
Thought Experiment
of falling bodies
The first explanation of the
pendulum motion was provided
within then new theory:
the Theory of Impetus
Albert of Saxony
Questions on the Four Books on the Heavens and
the World of Aristotle
According to this [theory], it would be said also that if the earth
were completely perforated, and through that hole a heavy body
were descending quite rapidly toward the center, then when the
center of gravity (medium gravitatis) of the descending body was at
the center of the world, that body would be moved on still further
[beyond the center] in the other direction, i.e., toward the heavens,
because of the impetus in it not yet corrupted.
And, in so ascending, when the impetus would be spent, it would
conversely descend. And in such a descent it would again acquire
unto itself a certain small impetus by which it would be moved
again beyond the center. When this impetus was spent, it would
descend again. And so it would be moved, oscillating (titubando)
about the center until there no longer would be any such impetus
in it, and then it would come to rest.
• Changing the paradigm of permanent motion
Pendulum presented an intermediate stage in the transition of the
perception of “natural motion” between circular and linear motions
Giovanni Batista Benedetti’s
(1530-1590) demonstration of
the instant nature of the rest in
a periodic motion
True motion
of a planet
P
Pedagogical potential:
Concepts
Instant velocity,
the split between
acceleration and velocity
a
p
Conception
Harmonic motion
Observer
b
Apparent motion
of the planet
Non-asked questions:
A
P
B
- Describe the movement at the terminal point(s) of pendulum
motion A, B
- Describe the movement at the top point P of a tossed body
- Characterize the sate of the body in points A, B, and P?
Pendulum and the concept of weight:
L
mg
T  3mg
T
Surprise:
mg
To appreciate this result change the setting to a swing
The result is independent on L
What do we feel on a swing?
What force do we perceive?
Why do people like swinging?
What do we feel at the
extreme points?
Swing can serve a setting to
introduce the understanding of
weight as a concept different
from the gravitational force
weight is a result of weighing
• Galileo’s approach to Pendulum
This approach yields
T L
Galileo’s law of the chords
Huygens’ formula for
mathematical pendulum
A
B
A
D
C
D
TAD=
TAD=TBD=TCD=…= 2 R/g
π
R/g
2
π
R/g
T1/4  2 R/g T1/4 
2
Galileo
Huygens
π
2
2
Educational benefit
Surprise
• The discrepancy between the circular and linear paths is essential.
It is regardless how close the chord approaches the line.
L
T  2π
g
gπ
2
T L
2
Is this an easy way to define units?
T1 sec L  1m
Foucault pendulum
What does it prove?
Common view:
Earth’s rotation
Common misconception: absolute motion
Conclusion
• The common account of pendulum is formal and instrumental
• The conceptual knowledge of pendulum is often ignored and this
significantly impoverishes students’ knowledge of physics
• Cultural issues (history of physics, interdisciplinary aspects) both
beneficial and enjoying are ignored
• The chance to learn about the nature of science (philosophical
aspects) is missed
‫תודה רבה‬