DETERMINATION OF MODULUS OF RIGIDITY BY MAXWELL'S NEEDLE
1. AIM
To determine the modulus of rigidity of the material of a given wire using Maxwell's needle apparatus.
2. APPARATUS USED
- Maxwell's needle apparatus
- Stop watch
- Meter scale
- Vernier caliper
- Screw gauge
- Set of weights (small masses)
- Wire sample (brass/steel/copper)
- Spirit level
3. DIAGRAM
4. THEORY
Maxwell's needle experiment is designed to determine the modulus of rigidity of the material of a wire by observing torsional oscillations. When a cylindrical bar (Maxwell's needle) is suspended by the wire under test and set into torsional oscillations, the time period of these oscillations is related to the modulus of rigidity of the wire material.
The modulus of rigidity (η) is defined as the ratio of shear stress to shear strain within the elastic limit. It characterizes how a material responds to shear stress.
When the Maxwell's needle is twisted and released, it executes torsional oscillations. The time period of these oscillations depends on:
- The moment of inertia (I) of the Maxwell's needle
- The torsional constant (C) of the wire
The torsional constant C is related to the modulus of rigidity η by:
Where:
- r is the radius of the wire
- L is the length of the wire
The relationship between the time period of oscillation (T), moment of inertia (I), and torsional constant (C) is:
In Maxwell's needle apparatus, we can determine the moment of inertia in two ways:
- By direct calculation based on the dimensions and mass of the needle
- By using a comparative method
In this experiment, we will use the comparative method where we determine the moment of inertia by observing the time period with and without additional masses placed at known distances from the axis of rotation.
5. FORMULA
The modulus of rigidity (η) is given by:
Where:
- η = Modulus of rigidity of the wire material (N/m²)
- L = Length of the wire (m)
- I = Moment of inertia of the Maxwell's needle (kg·m²)
- T = Time period of oscillation (s)
- r = Radius of the wire (m)
The moment of inertia is calculated using:
Where:
- M = Mass of each additional weight (kg)
- R = Distance of additional weights from the axis of rotation (m)
- T₁ = Time period with additional weights (s)
- T = Time period without additional weights (s)
6. PROCEDURE
- Ensure the Maxwell's needle apparatus is securely fixed to the support.
- Check that the wire is properly clamped at the top and attached to the center of the Maxwell's needle at the bottom.
- Use a spirit level to ensure the horizontal bar is perfectly horizontal.
- Using a screw gauge, measure the diameter of the wire at several positions and calculate the average radius (r).
- Using a meter scale, measure the effective length (L) of the wire from the fixed support to the point of attachment on the Maxwell's needle.
- Gently twist the Maxwell's needle through a small angle (not exceeding 5°) and release.
- Start the stopwatch when the needle passes its equilibrium position.
- Count 20 complete oscillations and record the total time.
- Divide this time by 20 to get the time period (T).
- Repeat this process three times and take the average value of T.
- Attach equal masses (M) symmetrically at both ends of the Maxwell's needle at a known distance (R) from the axis of rotation.
- Repeat step 3 to determine the new time period (T₁).
- Repeat with different positions of weights or different weights if required.
- Using the formula I = MR²(T₁²-T²)/(T²), calculate the moment of inertia of the Maxwell's needle.
- Using the formula η = 8πLI/(T²r⁴), calculate the modulus of rigidity of the wire material.
7. OBSERVATION TABLE
A. Measurement of wire dimensions:
| Parameter | Reading 1 | Reading 2 | Reading 3 | Average | SI Unit |
|---|---|---|---|---|---|
| Wire diameter | m | ||||
| Wire radius (r) | m | ||||
| Wire length (L) | m |
B. Time period without additional weights:
| Observation | Number of oscillations | Total time (s) | Time period T (s) |
|---|---|---|---|
| 1 | 20 | ||
| 2 | 20 | ||
| 3 | 20 | ||
| Average time period (T) | |||
C. Time period with additional weights:
Mass of each additional weight (M) = _____ kg
Distance from axis (R) = _____ m
| Observation | Number of oscillations | Total time (s) | Time period T₁ (s) |
|---|---|---|---|
| 1 | 20 | ||
| 2 | 20 | ||
| 3 | 20 | ||
| Average time period (T₁) | |||
8. CALCULATIONS
1. Calculate the moment of inertia (I):
Substitute the values:
- M = _____ kg
- R = _____ m
- T = _____ s
- T₁ = _____ s
I = _____ kg·m²
2. Calculate the modulus of rigidity (η):
Substitute the values:
- L = _____ m
- I = _____ kg·m²
- T = _____ s
- r = _____ m
η = _____ N/m² (or Pascal)
9. RESULT
The modulus of rigidity of the material of the given wire is _____ × 10¹⁰ N/m² (or _____ GPa).
10. PRECAUTIONS
11. VIVA VOCE QUESTIONS
Q: What is the modulus of rigidity?
A: The modulus of rigidity (or shear modulus) is the ratio of shear stress to shear strain within the elastic limit of a material. It measures the resistance of a material to shear deformation.
Q: Why is the Maxwell's needle experiment more accurate than a simple torsional pendulum?
A: The Maxwell's needle experiment uses a comparative method that eliminates errors due to unknown factors in the moment of inertia calculation and provides a more accurate measurement.
Q: How does temperature affect the modulus of rigidity?
A: Generally, the modulus of rigidity decreases with increasing temperature because thermal energy weakens the interatomic or intermolecular bonds in the material.
Q: Why should the oscillations be kept small in this experiment?
A: Small oscillations ensure that the torsional deformation remains within the elastic limit of the material and that the relationship between torque and angular displacement remains linear.
Q: What is the typical range of values for the modulus of rigidity for common metals?
A: For common metals, the modulus of rigidity typically ranges from 20 GPa (for aluminum) to 80 GPa (for steel).
Q: How does the modulus of rigidity relate to other elastic constants?
A: The modulus of rigidity (G or η) relates to Young's modulus (E) and Poisson's ratio (ν) by the equation: G = E/[2(1+ν)].
Q: Why do we use a cylindrical bar instead of a flat disc in Maxwell's needle?
A: A cylindrical bar with weights at its ends provides a larger and more easily adjustable moment of inertia, which improves the sensitivity and accuracy of the experiment.
Q: What would happen if we used a wire with a larger diameter?
A: A wire with a larger diameter would have a greater torsional constant, resulting in a shorter time period of oscillation for the same Maxwell's needle.
Q: How would you identify systematic errors in this experiment?
A: Systematic errors could be identified by repeating the experiment with different weights and distances, changing the wire length, or comparing results with standard values for the material.
Q: Why is the formula for torsional constant C = (πηr⁴)/(2L)?
A: This formula is derived from the solution of the torsion equation for a cylindrical rod, where the torque is proportional to the fourth power of radius, inversely proportional to length, and directly proportional to the modulus of rigidity.