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Chapter 7: Problem 922

To what depth below the surface of sea should a rubber ball be taken as to decrease its volume by \(0.1 \%\) [Take : density of sea water \(=1000\left(\mathrm{~kg} / \mathrm{m}^{3}\right)\) Bulk modulus of rubber $=9 \times 10^{8}\left(\mathrm{~N} / \mathrm{m}^{2}\right)$, acceleration due to gravity \(\left.=10\left(\mathrm{~m} / \mathrm{s}^{2}\right)\right]\) (A) \(9 \mathrm{~m}\) (B) \(18 \mathrm{~m}\) (C) \(180 \mathrm{~m}\) (D) \(90 \mathrm{~m}\)

Short Answer

Expert verified
The depth below the surface of the sea required to decrease the volume of the rubber ball by 0.1% is \(h = 90 \ m\). The correct answer is (D).

Step by step solution

01

Understand the given information

We are given the following information: - Density of sea water (ρ) = 1000 kg/m³ - Bulk modulus of rubber (B) = 9 × 10⁸ N/m² - Acceleration due to gravity (g) = 10 m/s² Our goal is to find the depth (h) below the sea surface so that the volume of the rubber ball decreases by 0.1%.
02

Formulate the equation for the volume decrease

To find the depth, we use the formula for the decrease in volume as a result of a pressure increase: ΔV/V = -ΔP/B Here, ΔV/V is the percentage change in volume, which is 0.1% (or 0.001 as a decimal), ΔP is the change in pressure, and B is the bulk modulus of the rubber.
03

Calculate the change in pressure

To calculate the change in pressure (ΔP) due to gravity, we can use the formula: ΔP = ρgh Where: ρ is the density of sea water, g is the acceleration due to gravity, h is the depth below the sea surface.
04

Substitute the pressure change and volume ratio in the equation

Now, we can substitute ΔP = ρgh and the ratio \(ΔV/V = -0.001 \) in the equation from step 2: -0.001 = -ρgh/B
05

Solve for depth h

Rearrange the equation from step 4 to solve for the depth h: h = (0.001 * B) / (ρg) We can substitute the given values: h = (0.001 * 9 × 10⁸ N/m²) / (1000 kg/m³ × 10 m/s²) h = 90 m
06

Choose the correct answer

According to our calculations, the depth h must be 90 meters (m) for the volume of the rubber ball to decrease by 0.1%. Therefore, the correct answer is: (D) 90 m

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