How to Prove that a Flying Pendulum follows SHM

[Rosella Lin]

Hi,

I was wondering if someone could please help me to understand :

1) How can I prove that a Pendulum is following SHM?
2) Also, does being isochronous also mean that the pendulum is following SHM?
Thank You very much.

 

[nasu]

It actually doesn't. The SHM is a good model just for small oscillations.
But what do you mean by "prove"? To show that the SHM model is good enough by doing some experiment? Or theoretically, by writing the equation of motion?

 

[sophiecentaur]

1) How can I prove that a Pendulum is following SHM?

This is an example of how you can take the simplest situation and show that there's a linear relationship between variables ( near enough linear for small displacements) . For very small swings (up to about 5 degrees), the restoring force for a plumb bob on a string it proportional to the angle of the string from the vertical (displacement). That will give an 'equation of motion' for the bob which turns out to be a second order differential equation. (If you haven't got as far as Calculus then you will just have to take that as a fact) and the solution to the equation gives you a time / distance relationship that's sinusoidal. For a mass and an ideal spring, the motion is sinusoidal over a large range of displacements because Well made) springs tend to follow Hooke's Law (Force is proportional to extension)

 

[Rosella Lin]

Actually I am working on a particular type of pendulum called the Flying Pendulum. I did not find anything on the internet on the physics behind its working and so I am trying to prove that in order for a clock to function, the movement has to be isochronous! The pendulum should follow SHM. I know about the second differential equation with an angle of 20 degrees and I want to do a general conjecture kind of explanation by proving LHS = RHS that it follows SHM. But I am really confused how to do that!

Thanks for your help

 

[Nidum]

A Flying Pendulum clock has the mechanism shown in this VIDEO

You can make no assumption about the pendulum motion being SHM or indeed any easily defined motion .
It should be possible to analyse at least the 'wrapping around the post' part of the motion mathematically .
Analysing the complete motion may be much more difficult .

 

[Rosella Lin]

Thanks a lot Nidum but what do you mean analyze it mathematically?

 

[sophiecentaur]

Actually I am working on a particular type of pendulum called the Flying Pendulum

This style of oscillator will have a very low timing accuracy. Any oscillator has an uncertainty in its frequency which relates to the inherent loss (friction in all the possible places in the mechanism) and the way that the energy is supplied to it (escapement mechanism) at the end of each system. The flying pendulum works on the basis of loss of energy as the winding / unwinding of the string causes the string to drop and miss the retaining peg. This gives the oscillator a very low 'Q' factor, compared with a regular pendulum, for instance, which will remain swinging for several tens of cycles after the drive is removed. This is a fundamental of high quality sustained oscillators. (Quartz oscillators will have a Q of many thousands.)
The poor accuracy may not be relevant to you because the mechanism has a high novelty value and such a clock is very stylish. It would be possible, of course, to cheat a bit and include some electronics which could lock the mechanical oscillator to a quartz oscillator and give you both style and accuracy.

I am trying to prove that in order for a clock to function, the movement has to be isochronous!

That goes without saying because the oscillator in any clock has to be isochronous (by definition). The 'regularity' that's required in the definition of the term 'isochronous' can involve a waveform other than a sinusoid but the cycle has to repeat over some period of time if a clock is going to work. The analysis would be a fair bit harder than for a simple pendulum. The wind/unwind motion could probably be analysed but what happens when the string becomes fully wound and starts to unwind would be very hard; there's a (very) inelastic collision with between the bob and the pillar which will absorb an amount of energy that would be very hard to estimate. I guess you could estimate it if you measure the difference in the string angle before and after the string first contacts the pillar.

 

[Rosella Lin]

Thank you so so so so so much. Thats reallllllllllly helpful! :)

 

[sophiecentaur]

A pleasure!

 

[Katarina KS]

What did you find out in your research? It sounds interesting..