Question

If a brass sphere of radius 36 cm is submerged in a lake at a depth where the pressure is \( 10^7 \) Pa, then the change in the radius of the sphere is (Bulk modulus of brass = 60 GPa)

• A. \( 4 \times 10^{-2} \) cm
• B. \( 2 \times 10^{-3} \) cm
• C. \( 4 \times 10^{-3} \) cm
• D. \( 2 \times 10^{-2} \) cm

Hint:

For small changes, the fractional change in volume is three times the fractional change in linear dimension (radius). \( \frac{\Delta V}{V} = 3 \frac{\Delta r}{r} \).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:

When submerged, the sphere experiences uniform pressure \( P \), causing a volume compression. We relate pressure, bulk modulus (\( K \)), and volume strain (\( \Delta V/V \)). Then we relate volume strain to linear strain (\( \Delta r/r \)).
Step 2: Key Formula or Approach:

1. Bulk Modulus: \( K = \frac{P}{|\Delta V / V|} \implies \frac{\Delta V}{V} = \frac{P}{K} \). 2. Relation between Volume and Radius change: \( V = \frac{4}{3}\pi r^3 \implies \frac{\Delta V}{V} = 3 \frac{\Delta r}{r} \). 3. Combine: \( 3 \frac{\Delta r}{r} = \frac{P}{K} \implies \Delta r = \frac{P \cdot r}{3K} \).
Step 3: Detailed Explanation:

Given: Radius \( r = 36 \) cm. Pressure \( P = 10^7 \) Pa. Bulk Modulus \( K = 60 \) GPa = \( 60 \times 10^9 \) Pa. Substitute values into \( \Delta r = \frac{P \cdot r}{3K} \): \[ \Delta r = \frac{10^7 \times 36}{3 \times 60 \times 10^9} \] \[ \Delta r = \frac{36}{180} \times \frac{10^7}{10^9} \] \[ \Delta r = \frac{1}{5} \times 10^{-2} \] \[ \Delta r = 0.2 \times 10^{-2} \, \text{cm} \] \[ \Delta r = 2 \times 10^{-3} \, \text{cm} \] Wait, let's recheck the calculation. \( 36 / 180 = 0.2 \). \( 10^7 / 10^9 = 10^{-2} \). Result: \( 0.2 \times 10^{-2} = 2 \times 10^{-3} \) cm. This matches Option (B). However, the Answer Key provided in the prompt indicates Correct Answer is (D) \( 2 \times 10^{-2} \). Let's check the units again. \( P = 10^7 \) Pa. \( K = 60 \) GPa = \( 60 \times 10^9 \) Pa. \( r = 36 \) cm. Maybe pressure is \( 10^8 \) or \( r \) is different? No, sticking to text. Let's re-read the option mark in image. In the image for Q92, Option 2 is marked with a green check: \( 2 \times 10^{-3} \) cm. Option 4 is \( 2 \times 10^{-2} \) cm. The text extraction might have listed options in a specific order in the prompt block (e.g., A, B, C, D). Looking at the screenshot: Option 1: \( 4 \times 10^{-2} \) Option 2: \( 2 \times 10^{-3} \) (Green Check) Option 3: \( 4 \times 10^{-3} \) Option 4: \( 2 \times 10^{-2} \) So the correct answer is indeed Option 2: \( 2 \times 10^{-3} \) cm. The "Correct Answer" in the prompt text above said (D), but based on my calculation and the visual tick mark, it should be (B). I will follow the visual evidence and calculation.
Step 4: Final Answer:

The change in radius is \( 2 \times 10^{-3} \) cm.