Problem 932 - JEE Physics

Question

A material has poisson's ratio \(0.50\). If uniform rod of it suffers longitudinal strain of \(2 \times 10^{-3}\). Then what is percentage change in volume ? (A) \(0.6\) (B) \(0.4\) (C) \(0.2\) (D) 0

Step-by-Step Solution

Answer

The percentage change in volume is \(0.2\%\).

Step 1: Identify the known values

We know the following values: - Poisson's ratio, \(\nu = 0.50\) - Longitudinal strain, \(\epsilon_l = 2 \times 10^{-3}\)

Step 2: Calculate volumetric strain using the equation

Use the equation \(\epsilon_v = (1 - 2\nu) \epsilon_l\) to find the volumetric strain: \(\epsilon_v = (1 - 2 \cdot 0.50) \cdot (2 \times 10^{-3}) = (-1) \cdot (2 \times 10^{-3}) = -2 \times 10^{-3}\)

Step 3: Calculate percentage change in volume

To find the percentage change in volume, multiply the volumetric strain by 100: Percentage change in volume = \(\epsilon_v \cdot 100 = (-2 \times 10^{-3}) \cdot 100 = -0.2\%\)

Step 4: Find the correct answer

The correct answer is a percentage change in volume of -0.2% which corresponds to the option (C) \(0.2\). However, since the answer should not be negative, we can assume that the question was looking for the absolute value of the change. In this case, the answer is 0.2%.

Key Concepts

Longitudinal Strain

Longitudinal strain is a measure of how much a material stretches or compresses along its length when a force is applied. It describes the deformation in one direction, typically expressed as a ratio of the change in length to the original length. Given by the formula:

• \( \epsilon_l = \frac{\Delta L}{L} \)

Here, \( \Delta L \) represents the change in length and \( L \) is the original length. The longitudinal strain is a dimensionless quantity and signifies a relative change.

Understanding longitudinal strain is crucial for engineering applications, where knowing how a material behaves under stress can determine its suitability for a particular purpose. For instance, construction beams need specific strain characteristics to ensure they don't buckle or break under heavy loads. In the given exercise, a longitudinal strain of magnitude \( 2 \times 10^{-3} \) is provided, which means the material elongates or compresses by that fraction of its original length.

Volumetric Strain

Volumetric strain is the measure of change in volume a material undergoes when subjected to stress. It is crucial in scenarios where the overall volume alteration affects performance, such as in pressure vessels or containers.

The formula for volumetric strain (\( \epsilon_v \)) can be derived depending on the material's Poisson's ratio (\( u \)) and its longitudinal strain (\( \epsilon_l \)). The relationship is given by:

• \( \epsilon_v = (1 - 2u) \epsilon_l \)

Here, Poisson's ratio quantifies the material's tendency to expand perpendicular to the applied force. A Poisson's ratio of 0.5, as given in the exercise, implies that the material is incompressible, making volumetric strain negative. This indicates a reduction in volume when stretched in one direction.

For the exercise, using the formula above, when we substitute \( u = 0.50 \) and \( \epsilon_l = 2 \times 10^{-3} \), we can calculate \( \epsilon_v = -2 \times 10^{-3} \), showing that the volume shrinks due to the Poisson effect.

Percentage Change in Volume

The percentage change in volume reflects how much the volume of a material changes relative to its original volume, expressed as a percentage. This is particularly useful because it provides a quantitative measure of deformation that can easily be communicated and compared.

To calculate the percentage change in volume, the volumetric strain is multiplied by 100:

• Percentage change in volume = \( \epsilon_v \times 100 \)

For instance, if the volumetric strain is negative, it indicates a decrease in volume. In our exercise, with \( \epsilon_v = -2 \times 10^{-3} \), the calculation results in a volume decrease of 0.2%.

It’s important to note that in some contexts, only the magnitude (absolute value) of the change in volume is sought, implying that we are interested in size rather than direction of change. Therefore, the absolute value gives 0.2% as the final answer in these scenarios. Understanding the percentage change in volume is essential in many fields. For example, knowing how much a material might shrink or expand informs design decisions in architecture and manufacturing.