The curve shape of a bent ruler—analytical, numerical and experimental studies

Abstract

A ruler, where one of its ends is axially displaced a known distance, is studied using analytical, numerical and experimental methods. The exact nonlinear differential equation for the bent ruler is derived and solved numerically. This exact differential equation is then simplified to a fourth order differential equation using a small angle approximation. Using this equation together with appropriate boundary conditions, an approximate curve shape for the ruler is determined, which turned out to be a sine function with no dependence on material parameters or cross-sectional data. An expression for the length of the bent ruler is derived and using the constraint that this length is constant, an equation for determination of a relationship between the midpoint transverse displacement (deflection) and the known axial displacement is obtained. This equation is solved numerically. In addition a simple algebraic relationship between the midpoint deflection and the axial displacement is derived. A comparison between measured data for a bent ruler, the numerical solution of the exact differential equation, the numerical solution of the approximate equation and the algebraic expression is presented. It is found that the algebraic sine-shaped solution gives reasonable results for moderate axial displacements. The experimental investigations, the algebraical expressions and the numerical simulations can be useful in high school teaching and at undergraduate university level.