Determination of Young's Modulus of a Wire by Optical Lever Method

4. Theory

When a wire is subjected to a load, it undergoes elastic deformation along its length. This deformation follows Hooke's Law within the elastic limit, which states that the strain produced in the wire is directly proportional to the applied stress.

Young's Modulus (Y) is defined as the ratio of stress to strain within the elastic limit:

Where:

• F = Applied force
• A = Cross-sectional area of the wire
• ΔL = Change in length
• L = Original length of the wire

In the optical lever method, the small extension in the wire is amplified using an optical lever arrangement. When the wire extends due to loading, the mirror attached to it rotates through a small angle. This rotation causes the reflected light beam to move on the scale by a distance that is proportional to the extension of the wire. This amplification allows for more precise measurement of the small extensions.

The optical lever principle states that if a mirror rotates through an angle θ, the reflected ray will rotate through an angle 2θ. This principle is used to measure very small angular displacements with high precision.

5. Formula

The Young's Modulus (Y) of the material of the wire is given by:

Where:

• Y = Young's Modulus in N/m²
• m = Mass applied in kg
• g = Acceleration due to gravity (9.8 m/s²)
• L = Length of the wire in m
• r = Radius of the wire in m
• ΔL = Extension in the wire in m

In the optical lever method, the extension ΔL is related to the shift in the position of the light spot on the scale (d) by:

Where:

• d = Shift in the position of the light spot on the scale
• b = Distance from the mirror to the point of support of the wire
• D = Distance from the mirror to the scale

Substituting, the final expression for Young's Modulus becomes:

6. Procedure

1. Setup Preparation:
   • Mount the experimental setup on a rigid support.
   • Use the spirit level to ensure the setup is horizontal.
   • Fix the upper end of the wire firmly to the support and attach the weight hanger to the lower end.
2. Measurement of Wire Dimensions:
   • Measure the diameter of the wire at different positions using a screw gauge. Take at least 5 readings and calculate the average diameter.
   • Calculate the radius (r) from the average diameter.
   • Measure the length (L) of the wire between the fixed point and the point where the mirror is attached using a meter scale.
3. Optical Setup:
   • Adjust the position of the lamp, mirror, and scale to form an optical lever arrangement.
   • Ensure that the light from the lamp reflects off the mirror and forms a clear spot on the scale.
   • Measure the distance (D) from the mirror to the scale.
   • Measure the distance (b) from the mirror to the point of support of the wire.
4. Taking Readings:
   • Note the initial position of the light spot on the scale with no load.
   • Add weights to the hanger in steps (e.g., 100g, 200g, 300g, etc.).
   • For each weight, note the new position of the light spot on the scale after the system has stabilized.
   • Calculate the shift (d) in the position of the light spot for each load.
   • Repeat the observations by removing the weights in the same steps and note the corresponding scale readings.
5. Verification:
   • Plot a graph of load (mg) versus shift in position (d). The graph should be a straight line, confirming Hooke's Law.

7. Observation Table

Measurement of Wire Diameter:

S.No.	Screw Gauge Reading (mm)	Mean Diameter (mm)
1
2
3
4
5
Average =

Load vs. Scale Reading:

S.No.	Mass m (kg)	Force mg (N)	Scale Reading (cm)	Shift d (cm)
0	0	0		0
1
2
3
4
5

Constants:

• Length of the wire (L) = _____ m
• Radius of the wire (r) = _____ m
• Distance from mirror to scale (D) = _____ m
• Distance from mirror to point of support (b) = _____ m
• Acceleration due to gravity (g) = 9.8 m/s²

8. Calculations

1. Calculate the cross-sectional area of the wire:

   $$A = \pi r^2$$

2. For each observation, calculate the extension in the wire:

   $$\Delta L = \frac{d \times b}{2D}$$

3. Calculate Young's Modulus for each observation using:

   $$Y = \frac{mg \times L}{\pi r^2 \times \Delta L}$$

   or

   $$Y = \frac{mg \times L \times 2D}{\pi r^2 \times d \times b}$$

4. Find the average value of Young's Modulus.

5. Calculate the percentage error:

   $$\text{Percentage Error} = \left|\frac{\text{Experimental Value} - \text{Standard Value}}{\text{Standard Value}}\right| \times 100$$

9. Result

The Young's Modulus of the material of the given wire is _______ N/m² or _______ GPa.

10. Precautions

1. The wire should be straight, uniform, and free from kinks or twists.
2. The load should be applied gradually without jerks.
3. The wire should not be loaded beyond its elastic limit.
4. The optical arrangement should be stable and free from vibrations.
5. Parallax error should be avoided while taking readings.
6. The scale should be placed perpendicular to the reflected ray.
7. The extension should be measured only after the system has stabilized.
8. The wire should be free from dust and rust.
9. The room should be properly illuminated with minimal external light interference.
10. Ensure that the temperature remains constant throughout the experiment.

11. Viva Voice Questions

Q: What is Young's Modulus?

A: Young's Modulus is the ratio of stress to strain within the elastic limit. It measures the stiffness or elasticity of a material.

Q: Why is the optical lever method preferred for measuring Young's Modulus of wires?

A: The optical lever method amplifies small extensions in the wire, making it possible to measure very small changes in length with greater accuracy.

Q: What is Hooke's Law and how is it related to Young's Modulus?

A: Hooke's Law states that within the elastic limit, the strain in a material is directly proportional to the applied stress. Young's Modulus is the constant of proportionality in this relationship.

Q: What factors affect the Young's Modulus of a material?

A: Temperature, alloying elements, heat treatment, and microscopic structure of the material affect Young's Modulus.

Q: How does temperature affect Young's Modulus?

A: Generally, Young's Modulus decreases with an increase in temperature as thermal energy weakens the atomic bonds in the material.

Q: What is the elastic limit?

A: The elastic limit is the maximum stress a material can withstand without permanent deformation.

Q: What is the significance of the slope in the load vs. displacement graph?

A: The slope represents the stiffness of the wire and is directly related to Young's Modulus.

Q: How does the diameter of the wire affect the extension produced by a given load?

A: For a given load, the extension is inversely proportional to the square of the diameter of the wire.

Q: Why is it important to ensure that the wire is not loaded beyond its elastic limit?

A: Beyond the elastic limit, permanent deformation occurs, and Hooke's Law no longer applies, leading to inaccurate determination of Young's Modulus.

Q: How does Young's Modulus relate to the intermolecular forces in a material?

A: Materials with stronger intermolecular forces generally have higher Young's Modulus values as they resist deformation more effectively.

Q: What are the typical units of Young's Modulus?

A: Young's Modulus is typically expressed in Pascal (Pa), Newtons per square meter (N/m²), or Gigapascals (GPa).

Q: How does Young's Modulus differ for different materials?

A: Different materials have different Young's Modulus values. For example, steel has a higher Young's Modulus than rubber, indicating that steel is stiffer and less elastic.

Q: Why is the graph of load vs. displacement expected to be a straight line?

A: According to Hooke's Law, within the elastic limit, stress is directly proportional to strain, which means load is directly proportional to extension, resulting in a straight-line graph.

Q: How does the length of the wire affect the extension?

A: The extension of a wire is directly proportional to its original length. A longer wire will undergo more extension than a shorter wire for the same applied force.