The Cornu Effect

Andrew Mark Allen - 05370299 November 13, 2011

Abstract

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In this experiment the bulk properties of perspex were investigated, specifically determining values for Young’s Modulus and Poisson’s ratio, found to be TODO: and TODO: respectively. The two values were determined by measurements on the interference pattern formed by the reflection of monochromatic light from the top of the perspex beam and from the bottom of a glass plate placed on top of the perspex beam, while the perspex beam, placed centrally on two knife edges a short distance apart, had weights suspended from either end.

Introduction and Theory

TODO: REWRITE The aim of this experiment was to determine, using a method of interference (Newton’s Rings), Young’s Modulus and Poisson’s ratio for perspex.

Young’s Modulus

Suppose we have a block of some material, with a force applied to the ends of the block. In that case, the stress is defined as the force per unit area on the block. This force results in a deformation of the material, which is quantified by the strain, which is defined as the change per unit length of the block, i.e. the ratio of the change in length to the original length.

This leads us to the definition of Young’s Modulus, which is defined for some material to be the ratio of stress to strain,

Y = \frac{stress}{strain}

which can, in crude terms, be seen to describe the stiffness of the material, i.e. a high value of Young’s Modulus corresponds to a stiff material, e.g. steel, while a low value corresponds to an elastic one, e.g. rubber.

Suppose we have some masses bending a beam, as per

Poisson’s Ratio

Poisson’s ratio is defined as the ratio of lateral strain to longitudinal strain,

σ = \frac{lateral strain}{longitudinal strain}

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Newton’s Rings

Newton’s rings are the interference patterns

Experimental Method

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0.0.1 Experimental setup

Figure 1: Experimental setup for Cornu Method experiment

The experiment was setup as in Figure 1. We wish to find out what distance The theory states that when the beam is loaded as shown in the experimental setup, the beam is bent into an arc with longitudinal radius of curvature R_{1}. Furthermore, the cross section is also bent into an arc in the opposite direction, with a transverse radius of curvature R_{2}.

From Figure 2 and from using Pythagoras’ theorem, we have the formula (R_{1} − f)^{2} + x^{2} = R_{1}^{2}, where f = d − d_{0}. Expanding and rearranging, we have f^{2} − 2R_{1}f + x^{2} = 0, solving for f (and discarding the larger solution) gives us f = R_{1} − \sqrt{R_{1}^{2} − 4x^{2}}. Assuming x ≪ R, we can use the Taylor series and discard the later terms to get f = R_{1} − (R_{1} − \frac{1}{2} \frac{x^{2}}{\sqrt{R_{1}^{2}}} ) = \frac{x^{2}}{2R_{1}} . Thus it can be

seen that the distance d of the perspex beam from the glass plate at a given distance x from the centre is given by the equation d = d_{0} + \frac{x^{2}}{2R_{1}}

Figure 2: Longitudinal curvature of beam

Results and Analysis

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Discussion and Conclusions

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