A 3 m long wire of radius 3 mm shows an extension of 0.1 mm when loaded vertically by a mass of 50 kg in an experiment to determine Young’s modulus. The value of Young’s modulus of the wire as per this experiment is $P \times 10^{11} \, \text{Nm}^{-2}$, where the value of $P$ is: (Take $g = 3\pi \, \text{m/s}^2$)
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Solution Steps
1
List Given Parameters
Length ($L$) = 3 m
Radius ($r$) = $3 \, \text{mm} = 3 \times 10^{-3} \, \text{m}$
Extension ($\Delta L$) = $0.1 \, \text{mm} = 10^{-4} \, \text{m}$
Mass ($M$) = 50 kg
Acceleration due to gravity ($g$) = $3\pi \, \text{m/s}^2$
2
Formula for Young’s Modulus ($Y$)
Young’s modulus is given by the ratio of stress to strain:
$$Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L} = \frac{MgL}{A \cdot \Delta L}$$
where $A$ is the cross-sectional area $\pi r^2$.
3
Substitute Values
$$Y = \frac{50 \times (3\pi) \times 3}{\pi (3 \times 10^{-3})^2 \times 10^{-4}}$$
Simplifying the denominator: $\pi \times 9 \times 10^{-6} \times 10^{-4} = 9\pi \times 10^{-10}$
$$Y = \frac{450\pi}{9\pi \times 10^{-10}}$$
4
Calculate Numerical Value
Cancel $\pi$ and divide 450 by 9:
$$Y = 50 \times 10^{10} \, \text{N/m}^2$$
$$Y = 5 \times 10^{11} \, \text{N/m}^2$$
5
Find the value of P
The question states the modulus is $P \times 10^{11} \, \text{Nm}^{-2}$.
Comparing $5 \times 10^{11}$ with $P \times 10^{11}$:
Value of P: 5
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Theory
1. Elasticity and Hooke’s Law
Elasticity is the property of a body by virtue of which it tends to regain its original size and shape when the applied force is removed. Within the elastic limit, Hooke’s Law states that stress is directly proportional to strain. This relationship is fundamental to structural engineering and materials science. For longitudinal deformation, the proportionality constant is Young’s Modulus ($Y$). If a material has a high Young’s Modulus, it is considered stiff and requires more force to achieve a specific amount of deformation.
2. Stress and Strain Concepts
Longitudinal stress is defined as the restoring force per unit cross-sectional area ($F/A$), where $F$ is typically the weight of the suspended load ($Mg$). Longitudinal strain is the change in length per unit original length ($\Delta L / L$). It is a dimensionless quantity. In experiments like the one described, the wire is usually kept very thin to ensure measurable extension even with relatively small masses, allowing for a precise calculation of the material’s elastic properties.
3. Factors Influencing Young’s Modulus
Young’s modulus is an intrinsic property of the material, meaning it depends on the nature of the material and its temperature, but not on its geometric dimensions (length or radius). Whether you have a 1 m wire or a 10 m wire of the same material, $Y$ remains the same. However, as temperature increases, the intermolecular bonds generally weaken, usually leading to a decrease in the Young’s modulus. This property is vital for selecting materials for building bridges, skyscrapers, and mechanical parts.
4. Experimental Determination (Searle’s Method)
The scenario described in the problem is a simplified version of Searle’s apparatus experiment. In this setup, two wires of the same material and length are suspended. One wire (the experimental wire) is loaded with weights, while the other serves as a reference to compensate for temperature-induced changes in length. By measuring the extension $\Delta L$ using a spherometer or a vernier scale for different masses $M$, and plotting a graph, one can determine $Y$ with high accuracy.
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FAQs
1
Why did we take g as 3π?
This is a specific value provided in the problem to simplify the calculation, as it allows π to cancel out with the π from the area formula (πr²).
2
Does the radius need to be in meters?
Yes, for the standard SI unit (N/m²), all dimensions like length, radius, and extension must be converted to meters.
3
What is the difference between Stress and Pressure?
While both are Force/Area, pressure is an external force acting on a surface, while stress is an internal restoring force resisting deformation.
4
Is strain really dimensionless?
Yes, it is the ratio of two lengths (meters/meters), so it has no units.
5
What is the elastic limit?
It is the maximum stress a material can withstand such that it still returns to its original shape upon removing the load.
6
How does Y relate to Bulk Modulus?
They are related via Poisson’s ratio ($\sigma$) through the formula $Y = 3K(1 – 2\sigma)$.
7
Can Y be negative?
No, Young’s modulus must be positive as materials require energy/force to be stretched.
8
What if the wire was thicker?
The extension $\Delta L$ would be much smaller for the same mass, but the calculated $Y$ would remain the same.
9
What material might this wire be?
With $Y = 0.5 \times 10^{11} \, \text{Pa}$, it is likely a material like copper ($Y \approx 1.1 \times 10^{11}$) or aluminum ($Y \approx 0.7 \times 10^{11}$), though the values in exam problems are often adjusted for calculation ease.
10
What is P in the final answer?
P is just the numerical part before the power of 10. Since $Y = 5 \times 10^{11}$, $P = 5$.
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