- #1
Corneo
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Hi I am working on a certain homework problem and I would appreciate some hint or inputs.
A rope, of length L, is attached to the ceiling and struck from the bottom at t=0. The rope has negible stiffness, how long would it take for the wave to travel up the string and back down?
I have worked on the problem for a while and concluded that the velocity will vary because tension in the rope varies as you travel along the medium.
Tension, T, can be written as the distance from the bottom of the rope. That is T(x)=x \mu g, x is the distance measured from the bottom of the rope; and \mu=m/L is the linear mass density.
This is where I am lost. I think the wave equation is where I should start off, but not sure how to apply it to this problem
\frac {\partial ^2 y }{\partial t^2} = v^2 \frac {\partial ^2 y}{\partial x^2} = \frac {T(x)}{\mu} \frac {\partial ^2 y}{\partial x^2}= \frac {x \mu g}{\mu} \frac {\partial ^2 y}{\partial x^2} = x g \frac {\partial ^2 y}{\partial x^2}
Any hints or inputs would be appreciated.
A rope, of length L, is attached to the ceiling and struck from the bottom at t=0. The rope has negible stiffness, how long would it take for the wave to travel up the string and back down?
I have worked on the problem for a while and concluded that the velocity will vary because tension in the rope varies as you travel along the medium.
Tension, T, can be written as the distance from the bottom of the rope. That is T(x)=x \mu g, x is the distance measured from the bottom of the rope; and \mu=m/L is the linear mass density.
This is where I am lost. I think the wave equation is where I should start off, but not sure how to apply it to this problem
\frac {\partial ^2 y }{\partial t^2} = v^2 \frac {\partial ^2 y}{\partial x^2} = \frac {T(x)}{\mu} \frac {\partial ^2 y}{\partial x^2}= \frac {x \mu g}{\mu} \frac {\partial ^2 y}{\partial x^2} = x g \frac {\partial ^2 y}{\partial x^2}
Any hints or inputs would be appreciated.