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Take a rubber band of length $l$ and mark a point $(\cdot)$ on it at $x^{th}$ distance assuming the elastic rubber band to be along the $x$ axis beginning from $x=0$.

My main query is: when the band is stretched, if it gets stretched horizontally and proportionally such that every point on the band elongates equally, then it would mean that when the end point is elongated to $p^{th}$ position, then the $(\cdot)$ moves to the $\big(\frac{xp}{l}\big)^{th}$ coordinate (I am considering $\text {x-axis}$ and this goes on as it stretches proportionally.

But what about the fact when the rubber band doesn't expand uniformly? Then it's not proportional.

Mathejunior
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1 Answers1

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You can start with a simple model for an elastic line of unstretched length $L$ which is described by a elastic potential of the form $$ U=\frac{1}{2} \int_0^L k(x) \left(\frac{d u(x)}{d x} \right)^2 dx $$ where $u(x)$ is the displacement of a point of the line from its unperturbed position $x$ induced by the stretching. This give the equation (obtained by minimization of the potential energy as Euler-Lagrange equations for the lagrangian ${\cal L}=-U$) $$ \frac{d}{d x} \left( k(x) \frac{du}{dx} \right)=0 $$ This can be integrated easily, obtaining $$ u(x) = A \int_0^x \frac{dy}{k(y)} + B $$ where $A$ and $B$ are integration constants. Imposing the boundary conditions $$ u(0)=B=0 $$ and $$ u(L)=A\int_0^L \frac{dy}{k(y)}+B $$ we find $$ u(x) = \frac{u(L)}{\int_0^L \frac{dy}{k(y)}} \int_0^x \frac{dy}{k(y)} $$ which gives the displacement of each point as a function of its position, for a given displacement of the end point. If $k$ is constant the displacement is proportional to $x$ $$ u(x) = \frac{x}{L} u(L) $$ As an example of a different situation consider $k=k_0(1+\beta x)$. In this case $$ u(x) = \frac{u(L)}{\int_0^L \frac{dy}{1+\beta y}} \int_0^x \frac{dy}{1+\beta y} $$ which gives $$ u(x) = u(L) \frac{\log \left(1+\beta x\right)}{\log \left(1+\beta L\right)} $$ You see that the displacement of each point is proportional to the displacement of the end point, only the proportionality constant is position dependent.

GCLL
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