Cornu

DAVID WHYTE 08513872 8 APRIL 2010

1

THE CORNU METHOD

ABSTRACT

By observing the interference pattern when monochromatic light is incident on a glass plate

on a Perspex bar under bending stress, the radii of curvature in both directions were

calculated. The obtained values were used to calculate Young’s modulus for Perspex:

2.2 ± 0.1 GPa; and Poisson’s ratio for Perspex: 0.29 ± 0.02.

THEORY & EQUATIONS

Stress is defined as the force per unit cross-sectional area applied to a

material (F/A). When a stress is applied to any material, it responds by

deforming in the direction of the force, i.e. its length L increases by some

amount ∆L. The quantity ∆L/L is referred to as the axial strain. Young’s

modulus is the ratio of the stress to the strain, given in pascals (Pa), and is a

useful measure of how stiff a material is.

The material may also respond to a stress by deforming in a direction

perpendicular to the direction of the stress. The strain in this direction is

called the transverse strain. The ratio of transverse strain to axial strain is

called Poisson’s ratio.

In this experiment, Young’s modulus and Poisson’s ratio were calculated

for a Perspex bar.

The force was provided by two 1 kg

weights, causing the straight bar to be

bend in one direction with radius of

curvature R1 and in a perpendicular

direction with radius of curvature R2. So

the bar is deformed into a doubly-curved

surface called a hyperbolic paraboloid.

The equation describing such a surface

is:

� ��

2��

�

2��

for a surface which curves ‘upwards’

along the x-axis and ‘downwards’ along

the y-axis.

A

L

∆L

F

Hyperbolic paraboloid with contour

lines of constant height drawn


2

Replacing z with the distance between the glass plate and the bar, d, and setting x = y = 0

gives a value to the constant and allows the equation to be written:

��

��

�

�� 2� � ��

where d0 is the distance between the plate and the bar at x = y = 0.

Constructive interference is seen when d is an integer

multiple of the wavelength of the light, λ. For d =

constant, the above equation describes sets of

hyperbolae sharing two common asymptotes. At y = 0,

the equation reduces to �� 2 �� and at x = 0,

it becomes � � �� 2 ��. So by measuring the

distances of the points where the fringes cross the x and

y axes, the two radii of curvature can be calculated.

The ratio R1/R2 can also be estimated by observing the

angle between the asymptotes and the axis. Since the

asymptotes are given by � ��/��, the angle θ

marked in the diagram is equal to arctan��/��.

Equivalently, R1/R2 = cot² θ.

The bending moment of a bar of cross-sectional area A is given by ��

��, where Y is Young’s

modulus for the material and k is its radius of gyration (equal to �/√12 for a bar of

rectangular cross-section with dimensions a and b). Balancing this with the moment of the

weight applied gives:

� � !"��

#$� �12 !"��

%�&

Poisson’s ratio for a bar under a bending force is simply given by the ratio of the radii of

curvature: R1/R2.

So when the wavelength of the light is known and the distances of fringes are measured; the

radii of curvature, and hence Young’s modulus and Poisson’s ratio, can be easily calculated.

θ

Two sets of hyperbolae with common

asymptotes, i.e. the shape of the fringes


3

If a convex lens is used (approximated as an elliptical paraboloid: z = kx2 + ky2, k > 0) in

place of a glass plate, the interference fringes will take the form of concentric ellipses:

3d representation of the surfaces in contact

fringes generated by lens-bar combination

EXPERIMENTAL METHOD

The apparatus was set up as shown above. a and b were measured using a Vernier callipers.

Using the travelling microscope, values of y for successive fringes were measured. The bar

was then rotated by 90° so values of x could be measured.

Using a protractor, the angle between the x-axis and the asymptote of the hyperbolic fringes

was estimated.

The glass plate was then replaced with a convex lens and the shape of the fringes was

observed.

Travelling microscope

Glass slide

Perspex beam

Glass plate

Sodium lamp

1 kg weight

Knife edges


4

RESULTS & ANALYSIS

x2 was plotted against N and a linear fit was performed on the data. The value of the slope

gives R1λ = 0.68 ± 0.04.

y2 was plotted against N and a linear fit was again performed on the data, this time

giving R2λ = 2.29 ± 0.08.

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8

x2(m

m2 )

N

0

2

4

6

8

10

12

14

16

18

1 2 3 4 5 6 7 8

y2(m

m2 )

N


5

Using a value of 589 nm for the wavelength of sodium light, this gives the radii of curvature:

R1 = 1.15 ± 0.07 m, R2 = 3.89 ± 0.08 m

Using the protractor, the angle θ was approximated to be 60°.

When the glass plate is replaced with a convex lens, the fringes appear as concentric ellipses,

as predicted by the theory.

CONCLUSIONS

Young’s modulus for Perspex was calculated using the determined value of R1:

� �12 !"��

%�& �12 ' 1 ' 9.81 ' 0.138 ' 1.15

0.3884 ' 0.006& � �2.2 � 0.1� 0 101Pa

Poisson’s ratio for Perspex was calculated:

4 ��

��

1.153.9

� 0.29 � 0.02

The value of θ read from the protractor gives σ ≈ cot2(60°) = 1/3

These values are relatively close to values obtained from tables available online1:

• Y = 2.7 – 3.5 GPa

• σ = 0.3 – 0.4

1 http://www.roymech.co.uk/

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