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Question: The density \(\rho\) of water of bulk modulus B at a depth y in the ocean is related to the density ...

The density ρ\rho of water of bulk modulus B at a depth y in the ocean is related to the density at surface ρ0\rho_{0} by the relation

A

ρ=ρ0[1−ρ0gyB]\rho = \rho_{0}\left\lbrack 1 - \frac{\rho_{0}gy}{B} \right\rbrack

B

ρ=ρ0[1+ρ0gyB]\rho = \rho_{0}\left\lbrack 1 + \frac{\rho_{0}gy}{B} \right\rbrack

C

ρ=ρ0[1+Bρ0hgy]\rho = \rho_{0}\left\lbrack 1 + \frac{Β}{\rho_{0}hgy} \right\rbrack

D

ρ=ρ0[1−Bρ0gy]\rho = \rho_{0}\left\lbrack 1 - \frac{B}{\rho_{0}gy} \right\rbrack

Answer

ρ=ρ0[1+ρ0gyB]\rho = \rho_{0}\left\lbrack 1 + \frac{\rho_{0}gy}{B} \right\rbrack

Explanation

Solution

Bulk modulus,

B=−V0ΔpΔV⇒ΔV=−V0ΔpBB = - V_{0}\frac{\Delta p}{\Delta V} \Rightarrow \Delta V = - V_{0}\frac{\Delta p}{B}

⇒ V=V0[1−ΔpB]V = V_{0}\left\lbrack 1 - \frac{\Delta p}{B} \right\rbrack

∴ Density, ρ=ρ0[1−ΔpB]−1=ρ0[1+ΔpB]\rho = \rho_{0}\left\lbrack 1 - \frac{\Delta p}{B} \right\rbrack^{- 1} = \rho_{0}\left\lbrack 1 + \frac{\Delta p}{B} \right\rbrack

where,Δp=p−p0=hρ0g\Delta p = p - p_{0} = h\rho_{0}g

= pressure difference between depth and surface of ocean

∴ ρ=ρ0[1+ρ0gyB]\rho = \rho_{0}\left\lbrack 1 + \frac{\rho_{0}gy}{B} \right\rbrack (As h = y)