DETERMINATION OF ELASTIC CONSTANTS – CORNU’S METHOD

Aim: To determine the elastic constants of the given material by Cornu’s interference method.
Apparatus: A glass or perspex plate is placed symmetrically on two knife-edges to form a
beam. The weight hangers are suspended from the beam symmetrically at a distance ‘p’ from
the nearby knife-edge. Equal masses are place on the weight hangers. A convex lens (to form
elliptical fringes) or a plane glass plate (to form hyperbolic fringes) is place at the middle of
the beam. The air film formed between the beam and the lens/glass plate is illuminated by
means of the light from a sodium vapor lamp. The light is made to fall normally on the air
film with the help of a glass plate arranged at 45o as shown in the figure. The interference
fringes formed are observes by means of a traveling microscope.
Principle:- when the beam is bent under the action of a load it has a definite radius of
curvature in longitudinal and transverse directions which depend upon the geometry of the
beam and Young’s modulus of the material. The radius of curvature changes with the change
of the load. From the study of these changes, elastic constants of the material of the beam can
be determined. The changes in radii of curvature in two mutually perpendicular directions are
investigated by optical interference method by forming Newton’s rings (elliptical or
hyperbolic fringes). A combination of glass plate/Perspex plate and a convex lens of large
focal length (~100cm) is used to get elliptical fringes and if a plane glass plate is used instead
of the convex lens , hyperbolic fringes are formed. In both of these cases . it can be shown

that D  n ,where D is the difference in squares of diameters or fringe separation and

2 2

n is order of the fringe. When the test plate is bent uniformly it suffers a longitudinal
elongation and transverse compressional strain at the same time. Let equal masses m1 be

suspended from the weight hangers. p is the distance between weight hangers and nearer
knife edge. If R1
Is the longitudinal radius of curvature of the bent plate,
YI 1 m1 gp
m1 gp  and  (1)
R R1 YI
1
where Y is the Young’s modulus of the material and I is geometrical moment of inertia of the
beam. For another mass m 2 , if R 2 is the corresponding radius of curvature,

1 m2 gp
 (2)
R2 YI
1 1 m  m1 gp
(2)- (1)    2 (3)
R2 R1 YI

If x1m and x1(mk ) are the distances between the m and the (m  k ) pair of dark fringes
th th

respectively in the longitudinal direction with mass m1 , then,

x12m  4R m (4)
1

x12(mk )  4R (m  k ) (5)
1

where  is the wavelength of light from the sodium vapor lamp.

(5)-(4)  x1( mk )  x  4 R k

2 2
1m 1

1 4k
 (6)
R1  x 2  x 2 
 1 ( m  k ) 1m 

Similarly if x 2 m and x2( mk ) are the corresponding distances with mass m 2 , then,

1 4k
 (7)
R2  x 2  x 2 
 2(m  k ) 2m 

1 1  1 1 
(7)-(6)    4k  2  2 
 x2( mk )  x2 m x1( mk )  x1m 
2 2
R2 R1

  4k  2   1 
1 1
(8)
R2 R1
 1 1
where  1  and 2 =
x2
1( m k ) x2
1m x22( m k )  x22m

from (3) and (8),

m2  m1 gp  4k   1 
2
YI

Y 
m2  m1 gp (9)
4k  2   1 I
bd 3
If’ ‘b’ is the breadth and ‘d’ is the thickness of the plate, then , I = . Substituting in (9),
12

3m2  m1 gp
Y (10)
4bd 3 k  2   1 I
Hence Y can be calculated.
' '
If R1 and R2 are transverse radii of curvature of the bent plate with masses m1 and m 2 , then,

1 1  1 1 
  4 k  2  2 
(11)
R2' R1'  y 2 ( m k )  y 2 m y1( m k )  y1m 
2 2

= 4k  2   1  (12)

1 1
where  2 = and  1 =
y 2
2 ( m k ) y 2
2m y2
1( m k )  y12m
here, y1m , y1( mk ) , y 2 m and y2( mk ) are the corresponding distances in transverse direction

for m1 and m2. Now Poison’s ratio,

Transversestrain Changeof curvaturein Y direction
 = =
Longitudin
al strain Changeof curvaturein X direction
  1 1
 '  ' 
=
 R2 R1  =  2  1 (13)
 1 1  2  1
  
 R2 R1 
Y and σ can also be calculated by plotting a graph taking the order of fringes along the X-axis
and square of the distance between corresponding fringes along the Y-axis. For any load M,
we can plot two straight line graphs ( one for longitudinal distance and the other for transverse

distance). Let  1 and  1' be the angles made by the straight lines with the X-axis in
longitudinal and transverse directions for a load M1. Similarly for load M2, let  2 and  2' be
the transverse corresponding angles. Then the slope,
dy dx
 tan  1  1M  4 R1
dx dM
1 1
 4Cot 1 and  4Cot 2 (14)
R1 R2
1 1
Similarly, '
 4Cot 1
'
and '
 4Cot 2' (15)
R1 R2
1 1
Hence     4 Cot 2  Cot 1 
 R2 R1 
3M 2  M 1 gP
 From (3) Y  (16)
bd 3 Cot 2  Cot1 

 '  4 Cot 2'  Cot1' 

1 1
Also, '
R2 R1
Cot 2'  Cot1'
From (13),   (17)
Cot 2  Cot1
Rigidity modulus n and bulk modulus k are given by
 Y Y
n and k
21    31  2 
Procedure : The experimental arrangement is done as shown in figure. Equal masses (M1
each ) are placed on the weight hangers. Bright and dark fringes are observed through the
microscope. The microscope is moved parallel to the length of the test plate(ie. in the
longitudinal direction). The cross wire is made tangential to the 20th dark fringe on one side (
say left) in longitudinal direction. The microscope reading is taken. The cross wire is then
made tangential to the 18th, 16th, 14th etc. and 2nd dark fringes on the same side and the
corresponding readings are taken. Then cross wire is adjusted tangential to the 2nd, 4th,….20th
fringes on the other side (right) in succession and the readings are taken in each case. The
diameters  x m  of fringes are calculated by taking the difference in the readings on the left

and right sides. The square of the diameter( xm ) is calculated and the value of x1( m k )  x1m
2 2 2

are determined for a constant value of k (say k=10). Hence,  1 is calculated. Similar

observations are recorded for transverse direction and the y1( m  k )  y1m are found. Hence
2 2

 1 is also calculated. The experiment is repeated for different load M2 and the corresponding
 2 and  2 are calculated. The distance between weight hanger and nearer knife edge is
measured as P. The breadth of the beam is found out using vernier calipers and the thickness
is measured using screw gauge. Hence Young’s modulus (Y) and Posson’s ratio(  ) are
calculated. Graphs are plotted by taking order of the fringes along X-axis and squares of

distance along Y-axis. Slopes of the n vs x( m k )  xm graphs are calculated as tan  and
2 2

substituting values of Cot  in the equation for Y and  , these quantities are calculated. The
mean value of Y,  , n and k are calculated from the values calculated using the observed
readings and the values calculated from graphs.

Results.
Young’s modulus Y = N/m2
 Poisson’s ratio,  =
Rigidity modulus, n = N/m2
Bulk modulus, k. = N/m2

Telescope

S
Glass plate
at 45o

OBSERVATIONS
1.Transverse readings :
M= gms. P= cm.

Order of Microscope reading Diameter

the D(cm) D2(cm2) Dm2 k  Dm2 (cm2)
fringe
Left(cm) Right(cm)
 Mean Dmk  Dm =
2 2

1
= = m-2
Mean ( Dm2 k  Dm2 )

2. Transverse readings:
M= gms. P= cm.

Order of Microscope reading Diameter

the fringe D(cm) D2(cm2) Dm2 k  Dm2 (c
Left(cm) Right(cm) m2 )

Mean Dmk  Dm =
2 2

1
= = m-2
Mean ( Dm2 k  Dm2 )

Repeat the experiment for another M value.

Breadth of the plate (Using vernier calipers)

Least count =
M.S.R V.S.R Total(cm)
 Mean breadth (b)= m
Thickness of the plate(Using screw gauge)

Least count =
P.S.R H.S.R Total(cm)

Mean thickness (d) = m

Young’s modulus, Y =

Longitudinal Transverse
M gm M gm
D (m )
2

D (m )
2

Slope = tan
2

Slope = tan
2

Order (n)
Order (n)

tan = cot = tan = cot  =

1 1