- #1
Gravitino22
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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]
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
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]
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]