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202309191729 Pastime Exercise 004

The blogger claims no originality of his problem below.

Let there be a rubber band of constant mass m (an invariable) but of non-constant length l (a variable), whose relaxed length (when unstretched) is l=L. May it sustain longitudinal elongation \Delta l to any degree,

i.e., \min (l)=L\leqslant l< \infty =\max (l);

and withstand uni-directional tension T to any extent,

i.e., \min (T)=0\leqslant T< \infty =\max (T);

as and when its cross-sectional area approaches the limit \displaystyle{\lim_{l\to\infty}A=0}.

Let there also be some pencil(s) of rigid body and in diameter comparable to the thickness of a slack rubber band.

Let the rubber band obey Hooke’s law.

\mathbf{F}=-k\,\mathbf{x}

where F is the magnitude of restoring force, k the spring constant, and x the magnitude of displacement from the equilibrium position.

Let the pencils be held firmly in any positions as desired, to each of which applies whenever necessary some force |\mathbf{F}|\propto T in direction pointing away from the centre of rubber band.

For your information.

The process of parallel layers sliding past each other is known as shearing.

A pile of papers, a pack of cards with rectangular cross-section can be pushed to obtain a parallelogram cross-section. In such cases, the angle between the sides has changed, but all that has actually happened is some parallel sliding.

Byju’s on Shearing stress

Setup.

Figures \text{\scriptsize{NOT}} drawn to scale.

\begin{aligned} T_{n}\bigg( x_{n,\, 0}=\frac{L}{n}\bigg) & = 0 \\ T_{n}(x_{n,\, 0} +\Delta x_n) & = T_{n}\bigg( \frac{L+\Delta l}{n}\bigg) \propto \Delta l\\ \end{aligned}

Problem.

Discuss what resistive (/restrictive) force \mathbf{F} the rubber band exerts on itself when it is in a circle, i.e., n\to\infty, expanding (radially), i.e., l\to\infty; such that in any infinitesimal sections, the direction of \mathbf{F} is orthogonal to that of tension \mathbf{T}, i.e.,

- F\,\hat{\mathbf{r}}=\mathbf{F}\perp \mathbf{T}=\pm T\,\hat{\boldsymbol{\theta}}.

This problem is not to be attempted.