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Chapter 7: Problem 925

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) \((\Delta \mathrm{V} / \mathrm{V}) \alpha \beta^{-2}\)

Short Answer

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

Step by step solution

01

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.
02

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 (\(\Delta V/V\)): \[\Delta V/V = -\frac{P}{B}\]
03

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) \((\Delta V / V) \propto B^{-1}\).

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