Young's modulus by single cantilever method

Young's modulus is a method to find the elasticity of a given solid material. The present article gives the explanation how to perform the experiment to determine the young's modulus by the use of material in the form of cantilever. The single cantilever method is used here.

1. Young’s Modulus of given solid
Praveen N. Vaidya
SDM College of Engineering and Technology, Dharwad.

Aim: To determine young’s modulus of material of given single cantilever.
Apparatus: Single cantilever, Travelling Microscope, Vernier Calipers, Screw gauge.
Theory:
If a material changes its shape and size on application of external force, without changing its position then
force applied is called as the deforming force. If a material able to regain its original shape and size after
the removal of deforming force then material is called as Elastic material.
STRESS: The deforming force applied per unit area of the material is called Stress. If stress is applied
along the length of the wire shaped material then is called as longitudinal or tensile stress. If stress
applied on the surface of material so as to change its volume is called volume stress or comprehensive
stress. If stress applied produce angular deformation then stress is called shearing stress or tangential
stress.
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N/m
2

STRAIN: By the application of stress on a material, the change in its dimension may takes place.
Therefore, the ratio change in the dimension of the material to its original dimension is called as STRAIN.
For Ex. Due to application tensile stress change in length of the material takes place, then ratio of change
in length of the body to original length is called tensile strain. Similarly, by the application of volume
stress change in the volume of material takes place, then ratio of change in volume of the material to
original volume is called volume strain.
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Sl. NO. Tensile strain our longitudinal
stress
Comprehensive strain or
volume strain
Shear strain

=
∆�
�

l – Original length, Δl – change
in length w.r.t applied stress
=
∆??????
??????

V – Original Volume, ΔV
– change in Volume
=��� ??????
θ – Angle through which two
positions of material twisted
relative each other.

2. MODULUS OF ELASTICITY:
If the applied stress exceeds certain limit then, material body does not regain its original shape and size.
Therefore Maximum stress for which a material body regains its original shape and size is called as
Elastic limit.
Under the elastic limit, Stress applied is directly proportional the Strain is called as Hooks law.
Therefore, Stress α Stain or
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Or the ratio of Stress to Strain is constant is called as Modulus of Elasticity.
YOUNG’S MODULUS OF ELASTICITY:
The ratio of Tensil Stress (Normal stress) to Tesnsil strain is constant and called as Young’s Modulus.

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′
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∆�/�
=?????? or ??????=
??????x�
∆�x??????
E - ??????����
′
� ??????������, SI unit is N/m
2
, as strain is unit less quantity
SINGLE CANTILEVER :

The cantilever is any beam or bar. The Elasticity (namely Young’s Modulus) of a material can be found
by taking that in the form of beam of suitable dimension fixed at its two ends or one end keeping other
end free. The cantilever which fixed one end keeping other end free is called as Single cantilever.
Simple Bending

A straight bar of homogeneous material is subject to only a moment at one end and an equal and opposite
moment at the other end...

Fig. 1
Assumptions

The beam is symmetrical about Y-Y
The traverse plane sections remain plane and
normal to the longitudinal fibres after
bending (Bernoulli’s assumption)
The fixed relationship between stress and
strain (Young's Modulus)for the beam
material is the same for tension and
compression ( σ= E.e )

3. Consider two section very close together (AB and CD) as shown in fig. 1. After bending, the sections
will be at A′B′ and C′D′ and are no longer parallel. AC will have extended to A′C′ and BD will have
compressed to B′D′. The line EF will be located such that it will not change in length. This surface is
called neutral surface and its intersection with Z_Z is called the neutral axis
The development lines A'B' and C'D' are intersecting at a point O at an angle of θ radians and the radius
of E′F′ = R
Let y be the distance (E'G') of any layer H'G' originally parallel to EF.
Then H′G′/E'F' =(R+y)θ /R θ = (R+y)/R
And the strain e at layer H'G' =
e = (H'G'- HG) / HG = (H'G'- HG) / EF = [(R+y)θ - R θ] /R θ = y /R
The accepted relationship between stress and strain is σ = E.e Therefore
σ = E.e = E. y /R
σ / E = y / R
From the above the following important simple beam bending relationship results

A simple beam subject to bending generates a maximum stress at the surface furthest away from the
neutral axis. Then tensile stress on the surface CA is given by.
σmax = ymax. M / I
Deflection of Beams:
A beam is basically defined as one structural member used to bear the different loads. In structure, beam
helps to bear the load and one must have been noted that, there will not be any structure without beams.
Below is shown the arc of the neutral axis of a beam subject to bending. Beam is usually subjected with
vertical load, shear load and also sometimes with horizontal load. We must have to note it here that cross
section of a beam will be quite smaller as compared to its length.
There are different types of beam used to perform the experiment of elasticity of material one of them is
Single cantilever method

4. Fig. 2
In a similar manner if an expression for the bending moment is known then the slope and deflection can
be obtained at any point x by single and double integration of the relationship and applying suitable
constants of integration.

Cantilever beam

Cantilever beam is basically defined as a beam where one end of beam will be fixed and other end of
beam will be free as in fig. 3. Following figure, displayed here, indicates the cantilever beam AB with
length L.

Fig. 3
As we can see here that beam AB is fixed at one end i.e. at end A and free at other end i.e. at end B and
therefore above beam will be termed as cantilever beam.
Consider a cantilever beam (uniform section) with a single concentrated load at the end. At the fixed end
x = 0, dy = 0 , dy/dx = 0

For small angle dy/dx = tan θ = θ
The curvature of a beam is identified as δθ /δs = 1/R
In the figure δθ is small ; δx is practically = δs; i.e ds /dx =1


From this simple approximation the following relationships are
derived.

5. From the equilibrium balance on cantilever at the support there is a resisting moment - FL and a vertical
upward force F. At any point x along the beam there is a moment F(x - L) = Mx = EI d
2
y /dx
2


Or Iy
FL
E
A
3
3

substituting, F = Mg, Where M – is mass applied and g – acceleration due to gravity = 9.8 m/s
2

??????=
��
3
12
for the recangular beam, where b – breadth and d – thickness of the
beam respectively
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′
� ������� �� �??????���� ����??????����� ??????� �??????��� �?????? 3
3
3
12
bdy
MgL
E
A



Rearranging the term taking mutually varying terms separately we have 








Ay
M
mean
bd
gL
E
3
3
4

Procedure:
Determine the width (b) and thickness (d) of the cantilever (scale) using vernier calipers and Screw gauge
respectively.
Determine the length (L) of the cantilever, is given by distance between tip of nail at the free end and
point at fixed end.
The travelling microscope set perfectly horizontal and the cantilever kept in such a way that the telescope
focused in the tip of nail attached free end of cantilever.
Adjust the vertical knob so that the horizontal cross wire should touch the tip of nail. Note down the
reading on the vertical scale of travelling microscope using the principle of vernier.

6. Now add a 10g wt hanger right below the tip of nail that create depression on the cantilever, again adjust
the microscope so that horizontal cross wire touch the nail and take the reading.
Some more readings are taken by adding weights each of 10g to the weight hanger.
Now repeat the process taking reading by removing the weight one by one. Tabulate the reading and
determine the mean value of mass per unit depression (M/yA).
Determine the Young’s Modulus by using 








Ay
M
mean
bd
gL
E
3
3
4
Result: Young’s Modulus of the material of given cantilever is____________N/m
2

OBSERVATIONS:

To determine the Breadth (b) of cantilever beam using Vernier Calipers:

Least Count (L.C.)of vernier callipers=
Value of one Main Scale Division (MSD)
Total number of division on vernier scale
=


Tr. No.

MSR in ________

CVD

TR= MSR+(CVDXLC) in_________











Breadth of the cantilever beam = mean TR = _________
To determine the Thickness (d) of cantilever beam using Screw gauge:
Z.E. ________
Pitch of Screw=
Distance uncovered on pitch scale
number of Rotations given to screw head
=

7. Least Count (L.C.)of Screw Gauge=
Pitch
Total number of division on Head Scale
=


Tr. No.

PSR in ________

HSR

TR= PSR + (HSR – ZE) x LC in_________











Thickness of the cantilever beam = mean TR = _________
Note: above equation, if zero error is -3 then we get HSR – (- 3) = HSR + 3 and
zero error is +2 then we get HSR – (3) = HSR – 3

To determine the Breadth (b) of cantilever beam using Vernier Calipers:

Least Count (L.C.)of Travelling Microscope=
Value of one Main Scale Division (MSD)
Total number of division on vernier scale
=

Tr. No. Mass
added in
gm (M)
TM reading for Mean
of
readings
Depression
of beam
(yA)
M/ yA
Load increasing
in______
Load decreasing
in____
















mean M/ yA = ________________________
So, Young’s modulus is given by, 








Ay
M
mean
bd
gL
E
3
3
4