Determination of young’s modulus of a bar by Bending method

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Abstract 

The aim of this experiment is to determine young’s modulus of a bar by bending method. This experiment helps to investigate the relationship between the load, span, height and the deflection of a beam/bar that placed on two bearers and affected by a constructed load at the centre. It also helps to ascertain the coefficient the elasticity for stainless steel, brass and mild steel.
Using the formula,
The young’s modulus of the given bar is found to be 20.6×1011 dyne/cm^2 with an error of 27.17%.

Introduction: [1]

Young’s modulus, numerical constant, named for the 18th-century English physician and physicist Thomas Young, that describes the elastic properties of a solid undergoing tension or compression in only one direction, as in the case of a metal rod that after being stretched or compressed lengthwise returns to its original length. Young’s modulus is a measure of the ability of a material to withstand changes in length when under lengthwise tension or compression. Sometimes referred to as the modulus of elasticity, Young’s modulus is equal to the longitudinal stress divided by the strain. Stress and strain may be described as follows in the case of a metal bar under tension.

If a metal bar of cross-sectional area A is pulled by a force F at each end, the bar stretches from its original length L0 to a new length Ln. (Simultaneously the cross section decreases.) The stress is the quotient of the tensile force divided by the cross-sectional area, or F/A. The strain or relative deformation is the change in length, Ln L0, divided by the original length, or (Ln L0)/L0. (Strain is dimensionless.) Thus Young’s modulus may be expressed mathematically as


Fig. 01 (Metal bar under tension increases in length and decreases in cross section EB inc)

This is a specific form of Hooke’s law of elasticity. The units of Young’s modulus in the English system are pounds per square inch (psi), and in the metric system newtons per square metre (N/m2). The value of Young’s modulus for aluminum is about 1.0 × 107 psi, or 7.0 × 1010 N/m2. The value for steel is about three times greater, which means that it takes three times as much force to stretch a steel bar the same amount as a similarly shaped aluminum bar.

Young’s modulus is meaningful only in the range in which the stress is proportional to the strain, and the material returns to its original dimensions when the external force is removed. As stresses increase, Young’s modulus may no longer remain constant but decrease, or the material may either flow, undergoing permanent deformation, or finally break.


When a metal bar under tension is elongated, its width is slightly diminished. This lateral shrinkage constitutes a transverse strain that is equal to the change in the width divided by the original width. The ratio of the transverse strain to the longitudinal strain is called Poisson’s ratio. The average value of Poisson’s ratio for steels is 0.28, and for aluminum alloys, 0.33. The volume of materials that have Poisson’s ratios less than 0.50 increase under longitudinal tension and decrease under longitudinal compression.

T heory: [2]


Here the given beam (metre scale) is supported symmetrically on two knife edges and loaded at it’s centre. The maximum depression is produced at it’s centre. Since the load is applied only at one point at the beam, the bending method is not uniform throughout the beam and the bending of the beam is called non uniform bending.

The following formula is used for the determination of Young’s modulus (Y) for a beam material.


Where, 
Y = young’s modulus of the material of the beam (dyne/cm2) Y0 = Depresional the cente of the beam (cm)
m = Mass suspended at the centre of the beam (gm)
l   = Distance between two knife edges (cm)
b = Breath of the beam (cm)
d = Depth of the rectangular beam (cm)

A pparatus:


Rectangular beam, Spherometer, suitable weights, a screw gauge, metre scale.

E xperimental diagram:

Fig. 02 (setting of apparatus)
A beam AB of the experimental material about a metre long and of uniform cross-section is mounted on two strong knife-edges K, K separated by a suitable distance. A small mm scale stands vertically on a knife-edges placed at the midpoint of the beam. A hook is attached to this knife- edges and supports a hunger on which weights are placed. A small low power microscope M fitted with cross-wires in the eye-piece and sliding along a vertical stand is placed at a suitable distance to read the scale. Sometimes a travelling microscope is used. It is possible to alter the distance between the supporting knife-edges. [3]

Experimental Data: 

(A) Length of the beam l1 between two supporting knife-edges 1) 90.3 cm 2) 90.3 cm 3) 90.3 cm. 
(B) Vernier constant of the slide calipers =
(C) Breadth (b) and depth (d) for the beam 
b: (i) 2.252 cm (ii) 2.52 cm (iii) 2.52 cm 
d: (i) 0.62 cm (ii) 0.616 cm (iii) 0.615 cm 

For screw gauge-
Pitch, P = 1mm, Total number of division, N = 100
Least count (L.C) = 

Table: Data for load vs depression:  


Calculation

From the graph (y0  vs m) when, load, m = 1400 gm
Depression, y0  = 0.3
Length of the beam, l = 90.3 cm 

Breadth of the beam, b = 2.52 cm 

Depth of the beam, d = 0.617 cm

Acceleration due to gravity, g = 980 cms-2

Young modulus,


P ercentage of error:



R esult


The value of young’s modulus of the bar measured by bending method is 14.22×1011 dyne/cm2 with an error of 27.17%


D iscussion:

The young’s modulus of the given bar is found to be 14.22×1011 dyne/cm2 27.17% of error is found in our experiment. For the error there may have been some reasons like-
(i)                 The knife-edges may not be rigid and fixed on rigid support.
(ii)               The knife-edges may not be at equal distance from the centre of bar/beam.
(iii)            The weight may not be placed and moved gently on the hanger.
(iv)             The load of beam may not be exceed the elastic limit of the beam.
(v)               To avoid the backless error, the circular scale of the screw gauge and spherometer may not be moved in one direction.
(vi)             After loading or moving weights, sometime may not be allowed before taking of the reading.

C onclusion: [5]

As a conclusion, the length of deflection is directly proportional to the magnitude of the force that applied on the surface of the beams; where the end of the beam are fixed or both of the ends are simply supported. The condition of young’s modulus for both ends using simple supported bearers is larger than the first one fixed end. The length of the beams is inversely proportional to the young’s modulus. Furthermore, stainless steel has higher value of young’s modulus compared to the rest and this makes it much stiffer than other specimen beams. Finally the width and height of a rectangular section beam is directly proportional to its moment of intertia.

R eferences:

[3]. Practical physics, Dr. Giasuddin ahmed and Md. Shahabuddin, page-51
[4]. lrrpublic.cli.det.nsw.edu.au/lrrSecure/Sites/Web/.../03_what_does_it_mean_03.htm

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