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Chapter 7: Problem 1027

The density \(\rho\) of coater of bulk modulus \(B\) at a depth \(y\) in the ocean is related to the density at surface \(\rho_{0}\) by the relation. (A) $\rho=\rho_{0}\left[1-\left\\{\left(\rho_{0} \mathrm{gy}\right\\} / \mathrm{B}\right\\}\right]$ (B) $\rho=\rho_{0}\left[1+\left\\{\left(\rho_{0} \mathrm{gy}\right\\} / \mathrm{B}\right\\}\right]$ (C) $\rho=\rho_{0}\left[1+\left\\{\left(\rho_{0} \mathrm{gyh}\right\\} / \mathrm{B}\right\\}\right]$ (D) $\rho=\rho_{0}\left[1-\left\\{\mathrm{B} /\left(\rho_{0} \mathrm{~g} \mathrm{y}\right\\}\right]\right.$

Short Answer

Expert verified
The correct expression for the density of coater \(\rho\) at a depth \(y\) in the ocean, given the bulk modulus \(B\) and the surface density \(\rho_{0}\), is \(\rho = \rho_{0}\left[1 + (\rho_{0}gy) / B\right]\). This is option (B). The remaining options either suggest that the density decreases with increasing depth, or include an undefined variable, which does not align with our understanding of pressure and density in the ocean.

Step by step solution

01

Examine option (A: ρ=ρ₀[1-(ρ₀gy)/B])

In this expression, it suggests that the density at depth y is equal to the surface density multiplied by a factor that's less than 1. This would mean that the density decreases with the increasing depth y. This contradicts our understanding of pressure and density in the ocean, where the density typically increases with depth. Hence, option (A) is incorrect. #Step 2: Examine option (B)#
02

Examine option (B: ρ=ρ₀[1+(ρ₀gy)/B])

In this expression, it suggests that the density at depth y is equal to the surface density multiplied by a factor greater than 1. This would mean that the density increases with increasing depth y. This agrees with our understanding of pressure and density in the ocean, where the density typically increases with depth. This could be the correct option, but let's check the other options too. #Step 3: Examine option (C)#
03

Examine option (C: ρ=ρ₀[1+(ρ₀gyh)/B])

In this expression, there's an additional variable "h" in the numerator. This variable is undefined and does not seem to have any physical meaning in the context of this problem. This would make this option incorrect. #Step 4: Examine option (D)#
04

Examine option (D: ρ=ρ₀[1-B/(ρ₀gy)])

In this expression, it suggests that the density at depth y is equal to the surface density multiplied by a factor that's less than 1. This would mean that the density decreases with the increasing depth y. This contradicts our understanding of pressure and density in the ocean, where the density typically increases with depth. Hence, option (D) is incorrect. #Step 5: Choose the correct option#
05

Choose the correct option

After examining all options, we can conclude that option (B) is the correct expression for the density of coater ρ at a depth y in the ocean, given the bulk modulus B and the surface density ρ₀: ρ = ρ₀[1 + (ρ₀gy)/B]

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