Kepler's second law and the pendulum?

[ollypenrice]

Gone a bit quiet on here so I have a question for you physicists. Is there not something to be seen in common between Kepler's second law (A line joining a planet and the Sun sweeps out equal areas during equal intervals of time) and Galileo's observations on the swing of a pendulum?

I'm not a physicist but it seems to me with my numbskull's grasp that the exchange of kinetic and potential energy applies in both cases.

If this is right then I'm slightly surprized that the Italian did not pick up on it. An historian would consider this speculation unhistorical but I think it is quite interesting - but only if there really IS something in common between the two discoveries. Your advice most welcome.

Olly

[brianb]

only if there really IS something in common between the two discoveries.

Both Kepler's Laws of planetary motion and Galileo's pendulum observations are consequences of Newtonian gravitation. Galileo had an incomplete grasp of gravitation as he would have been unable to deduce the inverse square law from which Kepler's Laws follow automatically. That's not to downplay the role of Kepler or Galileo, who did not have our model to work from.

This sort of thing often happens in science - we discover a phenomenon & only later find that it's a consequence of a more universal and possibly simpler law that we weren't aware of previously.

[ollypenrice]

Thanks Brian. I see that the inverse square law lies at the heart of the matter but what about the way potential and kinetic energy are exchanged in both systems (while conserving the same net eneregy?) Am I right here? The 'graphic' quality of Kepler's law - it is by its nature very visual - seems to be 'visually' so similar to the pendulum's behaviour that there was an intuitive clue in the air.

Olly

[brianb]

Since planets have no propulsion systems, potential & kinetic energy sum is constant (in fact for all orbits with the same semi major axis). If you understand this it also explains why reducing the total energy (retarding a satellite, making it drop into a lower orbit) actually speeds it up ... counterintuitive but this is exactly what the maths works out to, and precisely what the Gemini astronauts found in real life.

In actual fact - in the absence of any other disturbance - energy is NOT precisely conserved. The reason is that the planet (assuming it to be a constant uniform temperature) radiates more energy from the leading side than the trailing side; same number of photons of the same wavelength relative to the planet but shorter in wavelength on the leading side by Doppler shift. This imbalance in radiation acts as a (very weak) retro rocket, causing the orbit to shrink. Very very slowly when talking about planets, but the effect becomes considerable when talking about "planets" or other objects orbiting close to massive compact objects (white dwarfs, neutron stars & black holes).

As for the "graphical nature" - well, the only bit of Kepler that is really visual is the radius vector sweeping out equal areas in unit time. That sort of applies to Galilean pendulums too, but only so long as the amplitude of the swing is infinitesimal ... because the gravity always acts in the same direction, down, rather than through the pivot point.

[ollypenrice]

More thanks!

Olly

[Oldfort]

Kepler's second law comes from conservation of angular momentum, and is not directly related to gravity.

The motion of a pendulum needs the gravitional force, but it doesn't have to be an inverse square law - the typical derivation of the equation of motion of a pendulum assumes that the force of gravity is constant.

Kepler's first and third laws do depend on the inverse square law.

[brianb]

Kepler's second law comes from conservation of angular momentum, and is not directly related to gravity

Eh?

If the gravitational force varied with some other power of the distance, the orbit would not be an ellipse ... conservation of angular momentum of a body in elliptical motion is therefore a consequence of Newtonian gravitation.

Similarly - though a pendulum would work with a different distance law - the acceleration would be non uniform - the formula relating the speed of the bob and its displacement is a direct consequence of the rate of change of potential energy and the rate of change of kinetic energy having to be related, which does in fact boil down to the inverse square law when you do the calculus.

[Oldfort]

The orbit does not need to be elliptical for angular momentum to be conserved. The main requirement is that the force on the planet is in the same direction as the line from the sun to the planet ( a so called central force).

Newtonian gravity is an example of a central force, but an inverse cube or other power law would also result in Kepler's law. It would also apply to a particle on string where the string is slowly being wound in.

Pendulums don't conserve angular momentum, so represent a different phenomenon.

[ollypenrice]

Would it not be fair to say that there is something similar in the swing of the pendulm and the motion of a planet in an elliptical orbit? A speeding up and slowing down as kinetic energy is exchanged for potential? And both motions produce something which is non-variable arising from this exchange (the period of the swing and the swept area withn the orbit.)

I was interested in the idea of seeing or sensing a connection but I don't know how rigorous the idea of a connection really is.

Olly