Question:
Pressure of \(2 \times 10^6\,\text{Pa}\) causes volume decrease of \(0.1\%\) in a material. Bulk modulus is:
Pressure of \(2 \times 10^6\,\text{Pa}\) causes volume decrease of \(0.1\%\) in a material. Bulk modulus is:
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Always convert percentage changes into raw decimal fractions before substituting them into elastic moduli formulas ($0.1\% = 0.001 = 10^{-3}$). This avoids errors involving factors of $100$ in the final exponent.
Updated On: May 24, 2026
- $2 \times 10^9\text{ Pa}$
- $2 \times 10^{10}\text{ Pa}$
- $1 \times 10^{10}\text{ Pa}$
- $5 \times 10^9\text{ Pa}$
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
The problem asks to calculate the Bulk Modulus ($B$) of a material when a specified external pressure causes a given percentage decrease in its volume.
Step 2: Key Formula or Approach:
Bulk modulus ($B$) is defined as the ratio of volumetric stress (change in pressure, $\Delta P$) to volumetric strain (fractional change in volume, $\frac{\Delta V}{V}$):
\[ B = -\frac{\Delta P}{\frac{\Delta V}{V}} \] Since we are interested in the magnitude of the bulk modulus, we consider its absolute value:
\[ B = \frac{\Delta P}{\frac{|\Delta V|}{V}} \]
Step 3: Detailed Explanation:
• Let us write down the given values from the problem statement:
Applied excess pressure, $\Delta P = 2 \times 10^6\text{ Pa}$
Percentage decrease in volume $= 0.1\%$
• The percentage change in volume can be written as:
\[ \frac{|\Delta V|}{V} \times 100 = 0.1 \]
• From this, we find the fractional change in volume (volumetric strain):
\[ \frac{|\Delta V|}{V} = \frac{0.1}{100} = 10^{-3} \]
• Now, we substitute the values of volumetric stress ($\Delta P$) and volumetric strain into the Bulk Modulus equation:
\[ B = \frac{2 \times 10^6}{10^{-3}} \]
• Bringing the $10^{-3}$ term from the denominator to the numerator:
\[ B = 2 \times 10^6 \times 10^3 \] \[ B = 2 \times 10^9\text{ Pa} \]
• Therefore, the Bulk Modulus of the material is $2 \times 10^9\text{ Pa}$.
Step 4: Final Answer:
The Bulk Modulus of the given material is $2 \times 10^9\text{ Pa}$.
The problem asks to calculate the Bulk Modulus ($B$) of a material when a specified external pressure causes a given percentage decrease in its volume.
Step 2: Key Formula or Approach:
Bulk modulus ($B$) is defined as the ratio of volumetric stress (change in pressure, $\Delta P$) to volumetric strain (fractional change in volume, $\frac{\Delta V}{V}$):
\[ B = -\frac{\Delta P}{\frac{\Delta V}{V}} \] Since we are interested in the magnitude of the bulk modulus, we consider its absolute value:
\[ B = \frac{\Delta P}{\frac{|\Delta V|}{V}} \]
Step 3: Detailed Explanation:
• Let us write down the given values from the problem statement:
Applied excess pressure, $\Delta P = 2 \times 10^6\text{ Pa}$
Percentage decrease in volume $= 0.1\%$
• The percentage change in volume can be written as:
\[ \frac{|\Delta V|}{V} \times 100 = 0.1 \]
• From this, we find the fractional change in volume (volumetric strain):
\[ \frac{|\Delta V|}{V} = \frac{0.1}{100} = 10^{-3} \]
• Now, we substitute the values of volumetric stress ($\Delta P$) and volumetric strain into the Bulk Modulus equation:
\[ B = \frac{2 \times 10^6}{10^{-3}} \]
• Bringing the $10^{-3}$ term from the denominator to the numerator:
\[ B = 2 \times 10^6 \times 10^3 \] \[ B = 2 \times 10^9\text{ Pa} \]
• Therefore, the Bulk Modulus of the material is $2 \times 10^9\text{ Pa}$.
Step 4: Final Answer:
The Bulk Modulus of the given material is $2 \times 10^9\text{ Pa}$.
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