Elastic Energy in a Bent Rod · IPhO 2012 #1

Physics · energy-work

Problem

A uniform rod of length

L

with circular cross-section of radius

a

and Young's modulus

E

is bent into a circular arc of radius

R

(

R \gg a

). The bending energy is

U = \frac{EI}{2}\int_0^L (1/R)^2\,ds

where

I = \pi a^4/4

. Find

U

in terms of

E, a, L, R