Question

Pressure of $2 \times 10^6$ Pa causes volume decrease of $0.1\%$ in a material. Bulk modulus is:

• A. $2 \times 10^9$ Pa
• B. $2 \times 10^{10}$ Pa
• C. $1 \times 10^{10}$ Pa
• D. $5 \times 10^9$ Pa

Hint:

When the volume decrease is given as a percentage, the strain is simply $\frac{\% \text{ change}}{100}$. If $\Delta V/V$ is $10^{-n}$, the exponent of $10$ in the result will increase by $n$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Bulk modulus (\(B\)) is one of the three primary moduli of elasticity. It measures a substance's resistance to uniform compression.
It is defined as the ratio of volumetric stress to volumetric strain.
Volumetric stress is simply the pressure applied to the surface of the object.
Volumetric strain is the fractional change in volume (\(\Delta V / V\)).
Since the volume decreases when pressure increases, the strain is technically negative, but we usually look at the magnitude for the modulus.
Step 2: Key Formula or Approach:
The formula for Bulk Modulus is:
\[ B = \frac{\Delta P}{-\Delta V / V} \]
Where \(\Delta P\) is the change in pressure and \(\Delta V / V\) is the volumetric strain.
Step 3: Detailed Explanation:
Given values from the question:
- Applied Pressure (\(\Delta P\)) = \(2 \times 10^6\) Pa.
- Percentage decrease in volume = \(0.1%\).
First, we must calculate the volumetric strain (\(\Delta V / V\)) from the percentage:
\[ \text{Strain} = \frac{% \text{ change}}{100} = \frac{0.1}{100} = 0.001 = 10^{-3} \]
Now, substitute these into the Bulk Modulus formula:
\[ B = \frac{2 \times 10^6}{10^{-3}} \]
Using the law of exponents \(\frac{1}{10^{-n}} = 10^n\):
\[ B = 2 \times 10^6 \times 10^3 \]
\[ B = 2 \times 10^9 \text{ Pa} \]
Step 4: Final Answer:
The bulk modulus of the material is \(2 \times 10^9\) Pa.