Velocity of wave along a rubber cord

  • #1
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Homework Statement



You have a rubber cord of relaxed length x. It be-
haves according to Hooke's law with a "spring con-
stant" equal to k. You then stretch the cord so it has
a new length equal to 2x. a) Show that a wave will
propagate along the cord with speed

v=[tex]\sqrt{\frac{2kx^{2}}{m}}[/tex]

b) You then stretch the cord further so that the cord's
length increases with speed v/3. Show that the wave
will propagate during the stretching with a speed that
is not constant:

v(t)=[tex]\sqrt{\frac{kx^{2}}{m}(1+t\sqrt{\frac{2k}{9m}})(2+t\sqrt{\frac{2k}{9m}})}[/tex]

Homework Equations



strings wave propagation speed: v=[tex]\sqrt{\frac{T}{u}}[/tex]

hookes law: F=-kx

Where T is tension and u is linear mass density

The Attempt at a Solution



I have part A down

My train of thought for part b is that if your length is changing at a constant rate of v/3 then so is thetension. The new tension would be given by
T(t)=k(vt/3 -2x)
and the linear mass density
u(t)=m/(vt/3 +2x)

i plugged those into the velocity equation but i didnt get the result....Iam sure i have to use differentials but iam not so good at that so if anyone can point me in teh right direction
Thanks :)![/QUOTE]
 

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