Waves on a taut string

[davieddy]

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 well-defined 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

[Flatlander]

Originally Posted by davieddy

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 well-defined 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

You're just stating the obvious.
Any more Krankies' jokes?

[davieddy]

Originally Posted by Flatlander

You're just stating the obvious.
Any more Krankies' jokes?

:)

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.

[cheesehead]

Originally Posted by davieddy

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.

Hmm ... In my frosh physics lab, we used a wire strung between the lab and another building. I seem to recall that one thing we did was to pluck it once to start a wave, then pluck it again to make a second wave cross the first one coming back (easy to see as we sighted along the wire's length). I think they canceled (approximately) when they met.

[davieddy]

Originally Posted by cheesehead

Hmm ... In my frosh physics lab, we used a wire strung between the lab and another building. I seem to recall that one thing we did was to pluck it once to start a wave, then pluck it again to make a second wave cross the first one coming back (easy to see as we sighted along the wire's length). I think they canceled (approximately) when they met.

Cancelled displacement-wise, but the velocity was there, and so
did you also witness their "miraculous" reincarnation after they had
passed each other?

PS the number of hours I spent animating f(x-ct) - f(-(x+ct)) (or something like it)
on a blackboard doesn't really bear thinking about!

[davieddy]

Originally Posted by Flatlander

You're just stating the obvious.
Any more Krankies' jokes?

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).

[Flatlander]

I realised some time after posting that my post looked ambiguous, almost aggresive. I apologize; I was just teasing you about your joke.

[cheesehead]

Originally Posted by davieddy

Cancelled displacement-wise, but the velocity was there, and so did you also witness their "miraculous" reincarnation after they had passed each other?

Don't remember.

[davieddy]

Originally Posted by Flatlander

I realised some time after posting that my post looked ambiguous, almost aggresive. I apologize; I was just teasing you about your joke.

I prefaced my post with a smily too.
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