Problem 911 - JEE Physics

Question

The work per unit volume to stretch the length by \(1 \%\) of a wire with cross- sectional area \(1 \mathrm{~mm}^{2}\) will be.......... \(\left[\mathrm{y}=9 \times 10^{11}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\right]\) (A) \(9 \times 10^{11} \mathrm{~J}\) (B) \(4.5 \times 10^{7} \mathrm{~J}\) (C) \(9 \times 10^{7} \mathrm{~J}\) (D) \(4.5 \times 10^{11} \mathrm{~J}\)

Step-by-Step Solution

Answer

The short answer to the question is: The work per unit volume required to stretch the length of the wire by 1% is \(9 \times10^{7} J/m^3\).

Step 1: Convert Area to Standard Unit

First, we need to convert the cross-sectional area of the wire (1 mm²) to the standard unit (m²). 1 mm equals 0.001 m, so to convert the area: \(A = 1 mm^2 = (0.001m)^2 = 0.000001m^2 = 1×10^{-6}m^2\)

Step 2: Find the Strain

We are given that the length of the wire is stretched by 1%. Strain is the ratio of the change in length to the original length: \(Strain = \frac{\Delta L}{L} = 1\% = \frac{1}{100}\)

Step 3: Find the Stress

Using Young's modulus formula, we can find the stress: \(Y = \frac{Stress}{Strain}\) \(Stress = Y \times Strain = 9 \times 10^{11} N/m^2 \times \frac{1}{100} = 9 \times 10^{9} N/m^2\)

Step 4: Find the Work Done

To calculate the work done, we can find the product of stress, area, and change in length (∆L): \(W = Stress \times A \times \Delta L = 9 \times 10^{9} N/m^2 \times 1 \times 10^{-6} m^2 \times L \times 1\%\)

Step 5: Find the Work per Unit Volume

The work done per unit volume of the wire is the ratio of the work done to the volume of the wire given by (\(A \times L\)): \(Work \, per \, unit \, volume = \frac{W}{A \times L} = \frac{9 \times 10^{9} N/m^2 × 1 × 10^{-6} m^2 × L \times 1\%}{1 × 10^{-6} m^2 × L} = 9 \times 10^{9} N/m^2 \times \frac{1}{100}\) \(Work \, per \, unit \, volume = 9 \times 10^{7} J/m^3\) Hence, the work per unit volume required to stretch the length of the wire by 1% is \(9 \times10^{7} J/m^3\) which corresponds to option (C).

Key Concepts

Strain

Strain is a measure of how much a material deforms under stress. It is a dimensionless quantity that expresses the ratio of the change in length to the original length of the material. This helps in understanding how a material changes shape when a force is applied.
In mathematical terms, strain can be expressed as:

• Strain = \( \frac{\Delta L}{L} \)
• Where \( \Delta L \) is the change in length and \( L \) is the original length.

Imagine pulling a rubber band. The extension or stretch in the rubber band compared to its original length is the strain. In this example, if the rubber band stretches by 1% more than its original length, the strain is \( \frac{1}{100} \) or 0.01.

Stress

Stress is the internal resistance offered by a material when it is subjected to external force. It quantifies the intensity of internal forces acting within a deformed object. When stress occurs, it can lead to deformation depending on the material's properties. Stress is measured in units of force per area.
To calculate stress, Young's Modulus (a constant that defines the stiffness of a material) is often used. The formula connecting stress, Young's Modulus, and strain is:

• \( Y = \frac{\textrm{Stress}}{\textrm{Strain}} \)

Thus, for given values:

• \(\textrm{Stress} = Y \times \textrm{Strain} \)

Interestingly, stress can occur in different forms, like tensile stress, compressive stress, or shear stress, depending upon how the force is applied. In our example, the stretching of a wire is an example of tensile stress.

Work Done Per Unit Volume

Work done per unit volume refers to the energy required to deform material, given per cubic meter. It offers insight into how efficiently energy is used to achieve deformation. In mechanics, understanding this concept is crucial for materials used in designing and constructing structures.
Calculating the work done per unit volume involves finding the work done first and then dividing by the wire volume.

• Work done (W) can be represented as \( \textrm{Stress} \times \textrm{Area} \times \Delta L \)
• Volume is \( \textrm{Area} \times \textrm{Length} \)
• Thus, work per unit volume is \( \frac{W}{A \times L} \)

Through calculations, you can assess how much work and energy each unit volume of a material can endure to change its shape, providing insights into its strength and efficiency.

Cross-Sectional Area Conversion

Converting the cross-sectional area of a wire from millimeters squared to meters squared is essential in scientific calculations. Standard units ensure consistency and accuracy in physical equations, allowing them to be universally applicable.
When converting measurements:

• 1 mm = 0.001 m
• Thus, 1 mm² = (0.001 m)² = 0.000001 m² = 1 × 10^{-6} m²

This conversion is crucial when utilizing formulas involving area, such as when calculating stress or work done per unit volume. Using the correct units prevent errors in calculations and ensure that the results obtained are scientifically valid.