20090201, 09:52  #1 
"Lucan"
Dec 2006
England
14512_{8} Posts 
Waves on a taut string
At the second attempt, I succeeded in registering
with "Physics Forums" (where I probably belong) After two homework questions about waves on a string, and searching for the easiest way of explaining the formula for their speed, I recalled this (original from my teaching days although it must have been familiar long before): Consider the flexible string under tension T passing through a fixed wiggly tube at speed v. At any point, the tube's wiggly curve has a welldefined centre of curvature, and associated radius r. The acceleration of the string is v^2/r, and the force due to tension T is T*dl/r where dl is the length of an element of string. Both directed towards the centre of curvature. If T*dl/r = m*dl*v^2/r (m being mass/unit length) then there is no need for the tube! v^2 = T/m and there is no restriction on the waveform. ******************************************* And just for good measure, someone was asked to derive the moment of inertia of a rectangle about a corner by integration so I volunteered (after some previous help about double integration and the  axis theorem): Consider any plane shape and scale it up by two, keeping the mass per unit area the same. Each "element" has quadrupled in mass and doubled its distance from the C of M. It follows that the moment of inertia about an axis through the C of M normal to the plane has gone up by a factor of 16. For the a x b rectangle, let I be the moment of inertia about the C of M and J that about a corner. The  axis theorem gives J = I + m(a^2 + b^2)/4 Now consider the double sized rectangle formed from placing 4 of the originals together. We deduce that 16I = 4J = 4I + m(a^2 + b^2) ***************************** Anything to get rid of that inane "Beam up a Scottie" jest. David Last fiddled with by davieddy on 20090201 at 10:01 
20090201, 11:21  #2  
I quite division it
"Chris"
Feb 2005
England
31·67 Posts 
Quote:
Any more Krankies' jokes? Last fiddled with by Flatlander on 20090201 at 11:33 

20090201, 18:19  #3 
"Lucan"
Dec 2006
England
2·3·13·83 Posts 
:)
I thought making things obvious was a goal of teaching, so I shall optimistically treat this as a compliment. Although "any" waveform can travel in one direction on an ideal taut string, I would guess that the principle of superposition goes to pot, and if it met another wiggle coming in the opposite direction, they would not pass through each other unscathed like ideal waves do. Last fiddled with by davieddy on 20090201 at 18:20 
20090201, 18:39  #4  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}·3·641 Posts 
Quote:
Last fiddled with by cheesehead on 20090201 at 18:57 

20090201, 19:32  #5  
"Lucan"
Dec 2006
England
2·3·13·83 Posts 
Quote:
did you also witness their "miraculous" reincarnation after they had passed each other? PS the number of hours I spent animating f(xct)  f((x+ct)) (or something like it) on a blackboard doesn't really bear thinking about! Last fiddled with by davieddy on 20090201 at 20:23 

20090201, 20:37  #6 
"Lucan"
Dec 2006
England
2·3·13·83 Posts 
The "homework" section of Physics Forums is ludicrously busy,
and depressingly "formula" oriented. The recipient of my wisdom was having difficulty reconciling v = frequency*wavelength with v = SQR(T/m). Last fiddled with by davieddy on 20090201 at 20:44 
20090201, 22:19  #7 
I quite division it
"Chris"
Feb 2005
England
31×67 Posts 
I realised some time after posting that my post looked ambiguous, almost aggresive. I apologize; I was just teasing you about your joke.

20090202, 01:22  #8 
"Richard B. Woods"
Aug 2002
Wisconsin USA
7692_{10} Posts 

20090205, 05:10  #9  
"Lucan"
Dec 2006
England
194A_{16} Posts 
Quote:
Ever since our real time corresespondence about that spectacular Lunar Eclipse I have regarded you as a "friend". My post possibly deserved som putting down, and I can take it from you. I'd just like to add that a fellow "mentor" referred to my C of M derivation as "elegant reasoning". David 

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