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

[Bernard Cleyet <bernardcleyet@redshift.com>]

On 2011, Oct 08, , at 10:37, John Denker wrote:

> On 10/08/2011 09:29 AM, Bernard Cleyet wrote:
> > 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.
>
> That's not the point.
>
> The point is that changing the mass of a pendulum
> necessarily changes the restoring force in equal
> measure ... leaving the period unchanged, provided 
> the mass was added in such a way as to leave the length
> unchanged ... and indeed it would be equally easy to 
> add mass to the bottom (increasing the length) as to 
> add it to the top (decreasing the length).
>
> This stands in stark contrast to the mass on a 
> spring, where adding mass leaves the spring unchanged,
> and always lowers the frequency in proportion to the
> square root of the mass.
>
> > the equations are then the same.
>
> The equations are not the same.  With or without the 
> small-angle approximation, the physics is not the
> same.  The equivalence of gravitational mass and
> inertial mass is a deep principle of physics.  It
> applies to pendulums and not to springs.

You're quite correct.  I was confused by another's comment on the small angle regime -- the omegas squared are k/m and g/l .


> > Intuitively if added on the "down swing" the amplitude will be
> > increased and visa versa.
>
> That's not what my intuition says.  Basically it's
> just a mess, because if the added mass carries 
> momentum it disturbs the system, and if it doesn't
> carry momentum it disturbs the system.

The method my friend tried w/ a tower clock with bob mass of many kg was to drop a coin at ~ zero height above the bob, which has a flat top.  So inherently a mess, but ~ not too much in this practice.  


> > 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.  
>
> So get a more suitable core.  Ferrite cores work just fine up
> to megahertz frequencies, which really ought to be fast enough.
> And/or invert the mechanism so that turning the magnet *on*
> pulls a pin, thereby releasing the load.  Electric latches
> (including door latches) and electric bomb shackles have been
> around for a long time.  Back in the 1960s PSSC did a lot by
> holding stuff together with a thread, and then burning the
> thread.  Bottom line:  This part of the problem shouldn't be
> a stumbling block.

This involves winding a magnet, etc.  After some thought and before practice I turned to the above subject problem (driven pendulum).  I think all are convinced of the behaviour of adding mass to a free pendulum (ideally) to not bother experimenting further.


bc