Problem 1027 - JEE Physics

Question

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) \(\rho=\rho_{0}\left[1-\left\\{\mathrm{B} /\left(\rho_{0} \mathrm{~g} \mathrm{y}\right\\}\right]\right.\)

Step-by-Step Solution

Answer

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 1: 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)#

Step 2: 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)#

Step 3: 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)#

Step 4: 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#

Step 5: 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]

Key Concepts

Bulk Modulus

The bulk modulus, denoted as \( B \), is a material property that measures a substance's resistance to uniform compression. Think of it as a measure of how hard it is to compress a given volume of a substance: the higher the bulk modulus, the less compressible the material is.
In terms of equations, it is defined as \( B = -V \frac{dP}{dV} \), where \( V \) is the original volume, and \( \frac{dP}{dV} \) represents the change in pressure with respect to volume.
For oceans, understanding the bulk modulus helps in predicting how water density changes with depth. As we dive deeper, both pressure and density increase, but the water's high bulk modulus ensures these changes are gradual. This concept is crucial for calculating pressure and related phenomena in oceanography.

Ocean Pressure

Ocean pressure refers to the pressure exerted by the weight of water above a particular depth. It increases with depth due to the increasing weight of the water column. Pressure at any depth \( y \) can be calculated using the equation \( P = P_0 + \rho g y \), where \( P_0 \) is the pressure at the surface, \( \rho \) is water density, \( g \) is gravitational acceleration, and \( y \) represents depth.
This concept helps us understand why submarines need to be specially designed to withstand the immense pressures found at great depths. With increasing depth, pressure increases, compressing the water slightly, which in turn affects its density. This pressure-depth relationship is vital for various marine applications, from engineering to ecological studies.

Density at Depth

Density at depth relates to how the density of a fluid like seawater changes as you move deeper into the ocean. This change occurs because water is somewhat compressible, and pressure increases with depth, causing the molecules in water to pack more closely together.
As learned from the exercise, the relationship for density at depth \( y \) is expressed by the formula \( \rho = \rho_0[1+(\rho_0gy)/B] \). Here, \( \rho_0 \) is the initial surface density, \( g \) is the gravitational pull, \( y \) is the depth, and \( B \) is the bulk modulus.
This formula shows that density increases with depth, a concept important for understanding ocean currents, buoyancy, and the overall dynamics of underwater environments.

Surface Density

Surface density, noted as \( \rho_0 \), is the density of seawater measured at the ocean's surface. This value can vary due to temperature, salinity, and atmospheric pressure. For instance, saltier or colder water will have a higher surface density.
Surface density serves as a baseline for calculating how density changes when moving deeper into the ocean. In our relevant equations, \( \rho_0 \) helps us determine variations of density with depth by acting as a reference point. Understanding surface density is essential for a wide range of oceanographic and marine studies, including climatology and marine biology.
From predicting how sound travels underwater to the development of weather models, surface density plays a key role in various scientific applications.