Kepler's Laws, pendulae and a question.

[ollypenrice]

Historians these days disapprove of questions like this as 'unhistorical' but I'd still like to ponder this one, ideally with the help of a physicist.

1) Kepler's elegant second law notes that a line joining a planet and the sun sweeps out equal areas during equal intervals of time. (This might be unremarkable were the orbits circular but it is more thought provoking since they are elliptical.)

2) Galilleo noted that the period of the swing of a pendulum depended mainly on its length but was not affected by its amplitude.

Now if my physics is correct (by no means certain!) these phenomena arise from the fact that the sum of the kinetic energy and potential energy at any point in the swing of the bob or the orbit of the planet must be the same as at any other point. Moving slowly at aphelion the planet has less kinetic energy but more potential. The bob at its furthest point from rest has no kinetic energy, momentarily, since all is now potential. Although the pendulum loses energy due to friction it's period remains unaffected. And in Kepler, the swept area remains constant, a geometrical manifestation of the constant net energy of the system.

Given the 'visual' resemblance between a section of planetary orbit around perihelion or aphelion and the swing of a pendulum across the rest point, is it not odd that Gallileo didn't make some kind of intuitive connection? The presence of the two constants (the equal swept area and the equal pendulum period) seems to me to make this even more odd.

This question may be unhistorical but I think it's psychologically interesting. I know you can't have a notion of 'potential energy' without a notion of gravity but it's the visual resemblance that strikes me as compelling but overlooked.

Any thoughts on this, especially on the validity of the analogy that I'm seeing between pendulum swing and planetary orbit? (If this is invalid then my psychological speculations can stop immediately!)

Olly

[andrew s]

Olly - Kepler's second law actually come from the conservation of angular momentum rather than energy which is what is at play in a pendulum. The time keeping of a pendulum swing is only approximate and constant only for small amplitudes.

regards Andrew

[ollypenrice]

Yes, but doesn't a planet exchange kinetic energy for potential as its distance from the sun varies? In effect the conservation of angular momentum is describing the same situation from a different perspective, I'd have thought. The energy of the system is described as momentum but wouldn't it be equally valid to descirbe it as a combination of kinetic and potential energy?

This article from the net seems to gree with me:

The two very important conservation laws in nature: that of angular momentum (r x m x v = constant) and energy (here: kinetic + gravitational potential = constant) also explain Kepler's 2^nd Law. In the first case, as the planet's distance from the Sun (here called 'r') decreases, it's orbital speed must increase in order that its orbital angular momentum (r x m x v) remains constant. Just the opposite occurs as the planet's distance increases from the Sun. In the second case, as the planet's distance from the Sun decreases, so does its gravitational potential energy. Thus the planet's kinetic energy (proportional to the square of its orbital speed) must increase to compensate to keep the total energy = sum of kinetic + potential energies constant (conservation of energy). And of course, just the opposite applies as the planet's distance increases from the Sun - its potential energy increases, thus its kinetic energy (and so orbital speed) must decrease.

Olly

[andrew s]

Olly - Yes and no. They are two different laws not one of the same. If the conservation of angular momentum was not conserved then you would not get Kepler's second law but the pendulum would still have its constant beat. I accept that you need both conservation of energy (planet trading kinetic & potential) & momentum for Kepler’s second law.

As to how obvious the similarity between them is depends on your perspective and from the perspective of Gallileo I would not have expected him to make the link. We had to wait for Newton to link heaven & earth.

Regards Andrew

[acey]

A connection was indeed made between pendulums and planetary orbits, not by Galileo, but by Robert Hooke half a century later. He looked at pendulums where the bob is allowed to swing in a horizontal circle or ellipse. Public lecturers around that time or later would demonstrate planetary orbits using a circular pendulum with a pith ball as bob. By blowing on the ball they would make the path elliptical. Hooke's studies (which preceded Newton's Principia and possibly influenced it) are described in the following:

http://books.google....seminal&f=false

[ollypenrice]

A connection was indeed made between pendulums and planetary orbits, not by Galileo, but by Robert Hooke half a century later. He looked at pendulums where the bob is allowed to swing in a horizontal circle or ellipse. Public lecturers around that time or later would demonstrate planetary orbits using a circular pendulum with a pith ball as bob. By blowing on the ball they would make the path elliptical. Hooke's studies (which preceded Newton's Principia and possibly influenced it) are described in the following:

http://books.google....seminal&f=false

Ah, thanks, my curiosity seems somewhat vindicated, then. Although I've read Westfall, and Inwwood on Hooke, I'd not taken note of the use of pendulae as demonstrators of planetary motion.

The Newton-Hooke debate is a classic soap opera, of course. Had Faraday and Maxwell been characters like Hooke and Newton we'd doubtless have had a second instalment of intuition versus mathematics over electromagnetic radiation. Both Faraday and Maxwell were too well adjusted and downright nice for that, though.

Nevertheless, if intellects far meaner than Galilleo's would soon be seeing the connection between orbit and pendulum I'm still intrigued by what stood in Gallileo's way. What exact point of departure did he lack, I wonder.

Olly

[acey]

Nevertheless, if intellects far meaner than Galilleo's would soon be seeing the connection between orbit and pendulum I'm still intrigued by what stood in Gallileo's way. What exact point of departure did he lack, I wonder.

It seems that the stumbling block for Galileo was that he rejected Kepler's laws, and instead stuck with the Copernican model of circular orbits plus epicycles. Kepler suspected that his laws were due to a central force acting from the Sun but didn't have the means to treat the problem fully: Hooke and others (including Christopher Wren) shared this belief, and Newton solved the problem.

Galileo rejected the idea of a central force and to some extent stuck to older ideas (going back to Plato) about circular motion. See these links:

http://mathpages.com...68/kmath568.htm

http://books.google....l force&f=false

http://books.google.co.uk/books?id=0r68pggBSbgC&pg=PA137&lpg=PA137&dq=galileo's+rejection+of+kepler&source=bl&ots=UvLJqBrFNE&sig=oZtCp9MGoUM5eN6XXdgifPPVF-4&hl=en&sa=X&ei=M4wbUfnhLeXP0QXH-YCwDA&ved=0CDMQ6AEwAQ#v=onepage&q=galileo's%20rejection%20of%20kepler&f=false

[ollypenrice]

It seems that the stumbling block for Galileo was that he rejected Kepler's laws, and instead stuck with the Copernican model of circular orbits plus epicycles. Kepler suspected that his laws were due to a central force acting from the Sun but didn't have the means to treat the problem fully: Hooke and others (including Christopher Wren) shared this belief, and Newton solved the problem.

Galileo rejected the idea of a central force and to some extent stuck to older ideas (going back to Plato) about circular motion. See these links:

http://mathpages.com...68/kmath568.htm

http://books.google....l force&f=false

http://books.google.... kepler&f=false

I look forward to reading these links because I've long been confused by Gallileo's heliocentrism. His own description of a simple heliocentric model is at odds with the compexities of de Revolutionibus. It's as if he championed it without having read it. I wondered if he based his ideas on the Commentariolus rather than the final work.

Olly

[Tiki]

Olly, I am afraid that your article is misleading. With regards to planetary orbits, the conservation of angular momentum alone directly implies Kepler's 2nd Law. Kepler's 2nd law is a general property of central force motion and is not restricted to an inverse square law of force either. It clearly does not apply to a pendulum.

Areal velocity is a key concept when considering motion due to a central force.

[ollypenrice]

Olly, I am afraid that your article is misleading. With regards to planetary orbits, the conservation of angular momentum alone directly implies Kepler's 2nd Law. Kepler's 2nd law is a general property of central force motion and is not restricted to an inverse square law of force either. It clearly does not apply to a pendulum.

Areal velocity is a key concept when considering motion due to a central force.

OK, this is a modern understanding. But are you saying there is no equivalence between the exchange of kinetic and potential energy in a pendulum and a planetary orbit? I find this hard to believe but by all means convince me!

Olly

[nameunknown]

Easy way to visualise this (I think!): for a model pendulum with a small deflection the square of the period T is proportional to the length.

At first sight it does appear the "area swept out" is also proportional to the square of the length, that being some fraction of pi times r (length) squared.

Unfortunately that fraction can be anything from 0.5 to almost nothing - depending on through what angle the pendulum is swinging (in equal time)

The case of a conical pendulum (one swinging in a circle - see http://en.wikipedia.org/wiki/Conical_pendulum) is essentially the same as a normal one - the correction for the angle of the cone disappears for small angles, and it has the same period as a normal pendulum. As the circle gets smaller the period doesn't change, but the area swept out in the plane of the "orbit" shrinks away to nothing, meaning it cannot obey the equal area in equal times law in that plane either....

P

[Tiki]

OK, this is a modern understanding. But are you saying there is no equivalence between the exchange of kinetic and potential energy in a pendulum and a planetary orbit? I find this hard to believe but by all means convince me!

Olly

A planetary orbit is motion due to a central force, whereas a pendulum has a string (or something) which exerts a pull that when combined with gravity results in a non-central force.

We could also say that a planet on its orbit falls along a geodesic of spacetime, the pendulum bob is unable to follow a geodesic as it is constrained by a string.

The interplay between PE and KE lies at the heart of a great many physical systems.

[ollypenrice]

Thanks. I'm not suggesting that there is an exact analogy between a pendulum and a planetary orbit but that there might have been sufficient similarity to trigger an intuition. Maybe sharper minds than mine don't see the similarity because it's unsound!

Olly