Problem 925 - JEE Physics

Question

For a constant hydraulic stress on an object, the fractional change in the object volume \([\Delta \mathrm{V} / \mathrm{V}]\) and its bulk modulus (B) are related as............ (A) \((\Delta \mathrm{V} / \mathrm{V}) \alpha \beta\) (B) \((\Delta \mathrm{V} / \mathrm{V}) \alpha \beta^{-1}\) (C) \((\Delta \mathrm{V} / \mathrm{V}) \alpha \beta^{2}\) (D) \((\Delta \mathrm{V} / \mathrm{V}) \alpha \beta^{-2}\)

Step-by-Step Solution

Answer

The short answer is: (B) \((\Delta V / V) \propto B^{-1}\).

Step 1: Define the Bulk Modulus

The bulk modulus (B) is a measure of a substance's resistance to compression. It is defined as the ratio of the applied pressure (P) (also called hydraulic stress) to the volumetric strain, which is the fractional change in volume. Mathematically, it can be expressed as: \[B = -\frac{P}{\Delta V/V}\] Here, the negative sign indicates the volume compression (decreasing volume) due to the applied pressure.

Step 2: Solve for the Fractional Change in Volume

Now, let's rearrange the equation for the bulk modulus in order to solve for the fractional change in volume (\(\Delta V/V\)): \[\Delta V/V = -\frac{P}{B}\]

Step 3: Identify the Correct Relationship

Based on the relationship we derived in step 2, we can compare the given options and identify the correct one. The relationship we derived is: \[(\Delta V / V) \propto \frac{1}{B}\] Comparing this with the given options, we find that it matches option (B): \[(\Delta V / V) \propto B^{-1}\] So, the correct answer is (B) \((\Delta V / V) \propto B^{-1}\).

Key Concepts

Hydraulic Stress

Hydraulic stress refers to the pressure applied uniformly over the surface of a solid object. It can be thought of as the force exerted per unit area within the interior of a continuous medium. This stress is responsible for causing a change in volume of the material when pressure is applied.
Understanding hydraulic stress is important as it helps in analyzing how materials react under external pressures, such as in fluid environments or during material compression.

• Occurs when pressure is applied uniformly.
• Affects the entire volume of the object.
• Can cause the object to compress or expand, depending on the direction of the pressure.

In mathematical terms, hydraulic stress is often denoted by pressure (P), which plays a crucial role in calculating other properties like volumetric strain and bulk modulus.

Volumetric Strain

Volumetric strain is the measure of change in volume of a material due to applied stress. It shows how much the volume of a body changes in response to stress. Volumetric strain is a dimensionless quantity and is given by the formula:
\[ \text{Volumetric Strain} = \frac{\Delta V}{V}\]
where \(\Delta V\) is the change in volume and \(V\) is the original volume.
Volumetric strain helps in determining how susceptible a material is to deformation when subjected to stress. A higher volumetric strain implies the material deforms more easily.

• Calculated as the fractional change in volume.
• Helps in understanding material deformation under stress.
• Essential for determining material properties like the bulk modulus.

Fractional Change in Volume

The fractional change in volume is a specific application of volumetric strain. It assesses the ratio of the change in volume (\(\Delta V\)) to the original volume (\(V\)). This measurement is crucial in scenarios where volume alterations due to pressure are being analyzed.
This concept is particularly useful in fields like material science and engineering, where understanding how materials behave under varying pressure conditions is essential.

• Represents changes in the volume of a material.
• Critical in assessing material resistance to pressure changes.
• Integral in calculating the bulk modulus, a key property in material science.

Pressure-Volume Relationship

The pressure-volume relationship is essential for understanding how materials change under different pressure conditions. This relationship shows how pressure and volume interact, especially within the context of a material's bulk modulus.
When pressure increases, volume tends to decrease, and this inverse relationship is captured by the formula for bulk modulus: \[B = -\frac{P}{\Delta V/V}\]From this, it becomes clear that the fractional change in volume is inversely proportional to the bulk modulus:
\[\Delta V/V = -\frac{P}{B}\]

• Illustrates how pressure affects volume changes.
• Helps in predicting material behavior under stress.
• Useful in designing systems that must withstand specific pressure conditions.

Understanding this relationship allows engineers and scientists to manipulate variables like pressure to control volume changes in various applications.