To Determine Young's Modulus of Elasticity of the Material of a Given Wire

Searle's apparatus is used to determine Young's modulus of elasticity of the material of a given wire. Young's modulus is the property that informs us when an object or material will bend or break.

• The modulus is used for the measurement of the tensile or stiffness when the force is applied lengthwise.
• It is one of the most critical engineering tests. 
• The experiment is a form of measurement of a material's ability to stretch and distort. 
• The stiffness of a solid substance is measured by Young's modulus. 

When the externally applied force is withdrawn, it is only calculated for minor quantities of reversible elongation or compression that do not produce permanent deformation. 

• As a result, it's also known as elastic modulus. 
• It consists of two equal-length wires that are attached to a rigid support. 

Key Terms: To Determine Young's Modulus of Elasticity of the Material of a Given Wire, Stress-Strain Curve, Searle's Apparatus, Solid, Elastic modulus, Young's modulus, Force, Tensile Strength, Longitudinal Strain

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Aim

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The aim of the experiment is to determine Young’s modulus of elasticity of the material of a given wire.

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Materials Required

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The material required to determine Young’s modulus of elasticity of the material of a given wire are as follows:

• Searle’s apparatus
• Two long steel wires of the same length and diameter
• A meter scale
• A screw gauge
• Eight 0.5 kg slotted weights
• 1 kg hanger

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Theory

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The normal stress for a wire with length L and the radius r is loaded with weight Mg where l represents the increase in length, then normal stress is given as:

• Normal stress = Mg/ πr²
• Longitudinal strain = l/L
• Thus, Young’s modulus = Normal stress/ longitudinal strain Y = (Mg/ πr²)/(l/L) Y = MgL/ πr²l
• Y can be calculated as the values of L and r are known and l is found by known Mg value.

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Diagram

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The diagram to determine Young’s modulus of elasticity of the material of a given wire are as follows:

Diagram of Young Modulus 

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Measurement of Young's Modulus

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Young's modulus is a material attribute that is independent of its dimensions, such as length, diameter, and so on. Its value, however, is affected by the ambient temperature and pressure.

• To determine a material's Young's modulus, we usually employ Searle's approach.
• We can calculate such modulus using any shape because it is independent of the material's shape.
• A thin circular wire, in particular, meets our requirements.

In this procedure, a scale is used to measure the length L of the wire. A screw gauge is used to measure the diameter d of the wire, a Micrometer or Vernier scale is used to measure the length l of the wire, and F is the specified external force.

• The size of the strain produced in a given material is the same regardless of whether the stress is tensile or compressive.
• The ratio of tensile or compressive stress (σ) to longitudinal strain (ε) is defined or stated as Young's modulus.
• It is symbolized by the symbol 'Y.'

Y = σ ε

• Since strain is a dimensionless quantity, it is measured in N/m² or Pascal, the same unit as stress (Pa).

The Young's moduli and yield strengths of some of the materials are listed in the table below.

• A force of 2000 N is required to increase the length of a thin steel wire with a cross-sectional area of 0.1 cm².
• It increases the cross-sectional area by 0.1 per cent.

In aluminium, brass, and copper wires with the same cross-sectional area, the applied force required to create the same strain is 690 N, 900 N, and 1100 N, respectively.

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Determine Young’s Modulus of the Material of a Wire using Searle's Apparatus

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Searle's Apparatus is made comprised of two equal-length wires (control or reference wire and test wire) coupled to a sturdy support. At the opposite end, both the control and test wires are linked to a horizontal bar.

• This horizontal bar has a spirit level attached to it.
• This bar is now connected to the control wire by a hinge.
• When we raise the weight on the side of the test wire, it extends, causing the spirit level to tilt slightly.
• By adjusting the screw on the test wire side of a micrometer, we may alter the spirit level's tilt to any desired angle.
• To take the required readings, we return them to their horizontal position.

Micrometer and screw gauge are the same thing. It has a primary scale and a circular scale. The main scale and circle scale readings change as the screw is twisted to make the spirit level horizontal.

• The elongation l of the test wire is calculated using these readings.
• The control wire in a variant of Searle's apparatus holds a vernier scale that measures the test wire's extension.
• The slotted masses can change the force on the test wire.
• Let 'r' and 'L' stand for the experimental wire's initial radius and length, respectively.
• As a result, the wire's cross-sectional area is r².
• Let's call the mass that caused the wire to elongate or change in length L 'M.'
• As a result, the applied force is equal to Mg, where g denotes gravity's acceleration.
• The formula for calculating Young's modulus of the experimental wire's material is as follows:

Y =σ ε =Mg.l/πr² (change in l).

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Observations 

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The observations for determining Young’s modulus of elasticity of the material of a given wire are as follows:

• The Length of experimental wire AB, L = ….. cm = ……m
• The Measurement of diameter of the wire
• The Pitch of the screw gauge (p) = 0.1 cm
• Number of divisions on the circular scale = 100
• Least count of screw gauge (L.C) = 0.1/100 = 0.001 cm
• Zero error of screw gauge (e) = …….cm
• Zero error of screw gauge (e) = -e = ……cm

Diameter of the experimental wire is given as:

Measurement for extension of the wire

• Breaking stress for steel from a table, B = ….. Nm^-2
• Area of a cross-section of a wire, πr² = ….. cm² = …….m²
• Breaking load = Bπr² = …. N
• Bπr²/9.8 = …. Kg
• 1/3rd of the breaking load = ….kg
• Pitch of the spherometer screw, (p) = 0.1 cm
• Number of divisions in the disc = 100
• Least count of spherometer (LC) = 0.1/100 = 0.001 cm

Load and extension

The load and extension table is as follows:

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Calculations 

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The calculations for the experiment are as follows:

Table 1

Calculations from table 1 are as follows:

• Mean observed diameter of the wire,
• d₀ = d₁ + d₂ + … + d₁₀/ 10 = ….. Cm
• Mean corrected diameter of the wire,
• d = (d₀ + c) = ……cm = ……m
• Mean radius of wire,
• r = d/2 = ……m

Table 2

Calculations from table 2 are as follows:

• Mean extension for 2.5 kg load,
• l = l₁ + l₂ + l₃ + l₄/ 4 = ….. Cm = ….. m
• From formula,
• Y = MgL/ πr²l = 2.5 x 9.8 x l/ πr²l Nm^-2

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Results 

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The results of experiment to determine Young’s modulus of elasticity of the material of a given wire are as follows:

• As determined by Searle’s apparatus, the Young’s modulus for steel = …….. Nm^-2
• Straight-line graph between load and extension displays that stress ∝ strain, which verifies Hooke’s law.

Percentage Error

The percentage error is given as:

• Actual value of Y for steel =…… Nm^-2
• Difference in values =……… Nm^-2
• Percentage error = (difference in values)/(actual value) = …….%

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Things to Remember

• Searle's apparatus is used to determine Young's modulus of elasticity of the material of a given wire.
• Before beginning the experiment, kinks in the experimental wire should be eliminated.
• Both wires should be held in place by the same hard support.
• Weights with slots should be added and removed with care.
• The load should be gradually increased or lowered.
• After adding or withdrawing a weight, you should wait two minutes.
• A material's Young's modulus is crucial for practically everything around us, including buildings, bridges, automobiles, and more.

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