Pendulum as a culturally rich topic

to teach physics

Igal Galili

Science Teaching CenterThe Hebrew University of Jerusalem

2. Pendulum is used to test students’ understanding of force-motion relationship

Show/calculate the forces on the bob in…

Show/calculate the acceleration of the bob in…

Show/calculate the velocity of the bob in…

Questions asked in matriculation examination:

Calculate the period… g

l2πT

Pendulum is a traditional item in physics curriculum

1. Pendulum is used to practice measurements

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

Example:

Mathematical Pendulum is isochrornic (harmonic) only approximately

Pendulum can provide operational definition of time

Pendulum represents falling of bodies (Galileo)

Conical pendulum “explains” non-falling of bodies - imitates satellite (Hooke)

Features of the current presentation

Mathematical (mandatory) and physical (elected) pendulums are addressed.

Pendulum presents the case of vertical movement on a circular path

“When period is asked then the formula of Huygens is applicable”

Pendulum can measure time

Used: Missed:

Understanding of the force-motion relationship is tested

Pendulum

Model…

Object…

Theory…

What is it?

System…

What is it included for? What does it represent?

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.

Nucleus (center) – elements identifying the paradigms, concepts, principles, axioms, rules of knowledge production

nucleus

periphery

body

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

Physics as a Discipline-Culture

Holistic view on the subject of pendulum within the discipline-culture framework:

NucleusBody

Periphery

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)

Examples of materials which could be introduced

   1545-1607

del MonteGuidobaldo

Galileo

Newton

Collision of two views on the nature of science

Which one do we teach?

Two conceptions:Galileo’s versus Newton’s

Nucleus

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)

Time-measuringTime-meaning

Timekeeping

(instrumental)(operational)

(eventual)

Nucleus

• What makes pendulum moving? Aristotle: air

Philoponus: impetus

Galileo: gravity as a quality of bodies

Descartes: vortices of ether

Newton: Gravitational force

Einstein: Space Time curvature

Periphery

• Pendulum entered science in the Medieval European science of the 14th century

Jean Buridan (1290-1360)

Nicole Oresme (1320-1382)Albert of Saxony

(1316-1390)

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

Apparent motion of the planet

Observer

P

pbaInstant velocity ,

the split between acceleration and velocity

Conception

Harmonic motion

Concepts

Pedagogical potential:

- Describe the movement at the terminal point(s) of pendulum motion A, B

Non-asked questions: P

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?

L

Pendulum and the concept of weight:

mg

mg T3mgT

Surprise:

The result is independent on L

To appreciate this result -change the setting to a swing

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 LT

Galileo’s law of the chords

D

A

B

C

TAD=TBD=TCD=…= R/g2

A

Huygens’ formula for mathematical pendulum

D

R/g2

πTAD=

R/g2T1/4 R/g2

πT1/4

2

π2

HuygensGalileo

Educational benefit

• The discrepancy between the circular and linear paths is essential. It is regardless how close the chord approaches the line.

Surprise

g

L2πT

2πg LT2

Is this an easy way to define units?

T1 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

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