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Homework Statement
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A uniform rope of length L and negligible stiffness hangs from a solid fixture in the ceiling
The free lower end of the rope is struck sharply at time t=0. What is the time t it takes the resulting wave on the rope to travel to the ceiling, be reflected, and return to the lower end of the rope?
Express your answer in terms of L and constants such as g (the magnitude of the acceleration due to gravity), π, etc.
Homework Equations
v = sqrt (t/[itex] \mu [/itex] ) Where [itex] \mu [/itex] is mass per unit length.
The Attempt at a Solution
Starting off with the given equation:
v = [itex] \frac {T}{\mu} [/itex]
Using T = ma, I can replace T with [itex] \mu [/itex] z * g, where z is a length of string and g is the gravitational acceleration.
Now, equating a few equations:
v = [itex] \frac {dx} {dt} = \sqrt {gz} [/itex]
Solving for dt:
dt = [itex] \frac {dz} {\sqrt {gz} } [/itex]
From here, I integrate dt, so:
[itex] \int {dt} = \int { \frac {dz} {\sqrt {gz} } } [/itex]
Finally, I end with:
[itex] t = 2 \sqrt {gz} [/itex]
I just wanted someone to check my work to make sure I didn't make any silly mistakes in this. This was a long process and it was pretty hard, however, I am confused about one thing. Should "z" become L to represent the length of the entire string? Thank you for reading, and any help you can provide.