Question

The density ρ of coater of bulk modulus B at a depth y in the ocean is related to the density at surface ρ₀ by the relation. 

(A) ρ = ρ₀[1 – {(ρ₀gy} / B}] 

(B) ρ = ρ₀[1 + {(ρ₀gy} / B}] 

(C) ρ = ρ₀[1 + {(ρ₀gyh} / B}] 

(D) ρ = ρ₀[1 – {B / (ρ₀gy}]

Answer 1

Correct option: (B) ρ = ρ₀[1 + {(ρ₀gy} / B}]

Explanation:

ρ = density at depth Y.

ρ₀ = density at surface

B = bulk modulus

Change in density at depth y is

Δρ = [{– m } / {V₀}](ΔV / V₀)

Here ΔV = change in volume = V₀ – V

Now bulk modulus = B = [{– ΔP} / {ΔV / V₀}]

Hence

ΔV = (ΔP / B) ∙ V₀ 

∴ V₀ – V = [{V₀ ΔP} / B]

∴ V = V₀[1 – {ΔP / B}]        (1)

As density = [{mass} / {volume}]

We can write form (1)

ρ = ρ₀[1 – {ΔP / B}]^–1 

ρ = ρ₀[1 + {ΔP / B}]         (2)

Now P = h ∙ ρ ∙ g ------by definition

Here h = depth of ocean = y

∴ ΔP = g ∙ y ∙ ρ₀ 

∴ from (2)

ρ = ρ₀[1 + {(gyρ₀) / B}]

ρ = ρ₀[1 + {(ρ₀gy} / B}]