[Phys-l] Mass and SHO amplitude Was: Re: Student Misconceptions

[Bernard Cleyet <bernardcleyet@redshift.com>]

I thank all for replying to my request.  Since JD's reply is the most detailed I'll interleave his and where appropriate include others'.

On 2011, Oct 07, , at 09:33, John Denker wrote:

> On 10/07/2011 05:53 AM, chuck britton wrote:
>
> > If the mass miraculously increases ...
>
> 1) Memo from the keen-grasp-of-the-obvious department:
> I don't believe in miracles.  The mass cannot miraculously
> increase.  This would violate the laws of physics.  It would
> invalidate the equations of motion ... so figuring out what
> the subsequent motion would look like is quite impossible.
>
> As others have pointed out repeatedly, if you allow some
> mass to move across the boundary of the system, you need 
> to ask how it was done, and what else moves along with it:  
> how much energy, how much angular momentum, et cetera.

Right that was addressed by several as to when the mass is added.  I think the method was assumed with as little interference as possible.  Experimentally, for example to gently add the mass small compared to the existing bob's mass.  Intuitively if added on the "down swing" the amplitude will be increased and visa versa.**



> 2) Even if you did manage to change the mass of a pendulum,
> doing so is not analogous to changing the L in an LC
> oscillator, despite the claims made in the message that 
> started this sub-thread.
>
> The LC oscillator is analogous to a mass on a spring, in
> ways that a pendulum is not.

I'm not going to bother checking if I was explicit that this was an ideal (simple) pendulum wherein it is a linearized one.  One respondent did mention the small angle approximation.  So the equations are then the same.


> Feynman said "The same equations have the same solutions."
> However, the converse does not hold.  Just because you
> have the same solution (simple harmonic motion) does not
> mean the equations of motion are the same.

I don't think the solution of the nonlinear equation is simple harmonic.

Often not:


http://docs.google.com/viewer?a=v&q=cache:NUYVxvWsAZ4J:https://www.amherst.edu/media/view/79596/original/Simulating%252BNonlinear%252BOscillators.doc+nonlinear+oscillator+simple+harmonic+solution&hl=en&gl=us&pid=bl&srcid=ADGEEShD24hmHfzo5W4qTxxt8vZh6k289kItkodFZbFJ0yIz3fIY_qBn4G9Y_Jy5ybrQlHTLbpRSDz23qGqjVinTIatfDkSPlrv89mx9oaCWXiQdzPTGiJW1K3vAXPxvFt8UR93UbPhn&sig=AHIEtbQnP9xy-H00lHfWblwR0b7TDa11pw



> 3) As previously mentioned on multiple occasions, the
> theory of what happens when you change the L or the C

cut

> 4) It is even possible to account for the damping in an 
> LRC oscillator -- a physically-correct quantitative
> accounting -- which was first figured out relatively 
> recently.  A relatively accessible treatment can be 
> found at
>   http://ajp.aapt.org/resource/1/ajpias/v52/i12/p1099_s1


I though I mentioned the damping was the simplest mathematically, i.e. linear.


I realize now I was insufficiently explicit, i.e. too telegraphic.  I repeat:
------------------------------------------------
cut 

What happens when adding mass to a driven at resonance mechanical oscillator.  (spring or pendulum)  Note:  It, of course increases the "Q".

How about adding inductance in an LCR oscillator?  (L is the analog of mass.)

bc
-------------------------------------------------
All (but JD?) assumed an un-driven pendulum.  A clock pendulum is a fed-backed system wherein the drive is "in resonance", except transiently when disturbed.  Equilibrium obtains at a time depending on the Q.  So my question is, what is the change (if any) in the equilibrium amplitude?



** I spent several days in an attempt to drop a hanging mass from a bob.  My app. consisted of an EM powered by a solar cell.  when I shut off the lamp the weight dropped.  It was, of course unsuccessful, because of the retentivity of the EM's core.  It was at this point that I switched to an examination of the solution of the linearized in resonance driven pendulum.  More later.


bc