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Problem:

Consider a string with mass per unit length $\rho$ and length $L$. It is spun about one end, with angular velocity $\omega$ , such that the motion is in a plane (we neglect gravity).

Let $x$ denote the distance from the stationary end, and $u(t,x)$ be the vertical displacement at time $t$ at $x$. Assume that displace is small enough to allow for linearization.

Using the fact that centripal force exerted on a mass $m$ is a circle of radius $r$ is $F = m r^2 \omega^2$, I've shown that the tension is

$T(x) = \frac{1}{2} \rho \omega^2(L^2 - x^2)$.

Using Netwon's second law, I've shown that the motion of the string is modeled by

$\rho \frac{\partial^2 u}{\partial t^2} = \frac{\partial}{\partial x} (T \frac{\partial u}{\partial x})$

Question:

I am having trouble coming up with the boundary conditions for this problem.

One of the boundary conditions is $u(t,0) = 0$, since the end of the string is stationary. But what about the other end? Would it be $\frac{\partial u}{\partial x} |_{x=L} = 0$, since the vertical displacement cannot change with respect to $x$ past the end of the string?

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  • $\begingroup$ Your first problem is that "$u(t,x)$" does not describe the motion of the string if $x$ is "vertical". In an inertial coordinate system the string doesn't remain "vertical" because it is rotating! In a rotating coordinate system, the string won't necessarily stay straight, because of the Coriolis effect. Note, this isn't an answer given as a comment, because you haven't said what behaviour of the string you are trying to model. $\endgroup$ – alephzero Apr 12 at 22:52
  • $\begingroup$ @alephzero My apologies for being unfamiliar with this, as I don't know if this will answer your concerns. I am coming from a background of "pure" mathematics and this is an exercise for an "applied" math class. A lot of specific physical details in our models are neglected for sake of pedagogy. Also I forgot to mention another assumption, that the displacements are small. I am trying to model the 2-dimensional motion of the string. $\endgroup$ – SimonP Apr 13 at 14:24

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