When a large bubble rises from the bottom of a lake to the surface, its radius doubles. If atmospheric pressure is equal to the pressure of a column of water of height H, find the depth of the lake.
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Question

When a large bubble rises from the bottom of a lake to the surface, its radius doubles. If atmospheric pressure is equal to the pressure of a column of water of height H, find the depth of the lake.

G1 Physics · Accepted Answer

Step 1: The process is assumed isothermal, so Boyle's law applies: P₁V₁ = P₂V₂. Step 2: Since the radius doubles, V₂/V₁ = (2r/r)³ = 8. Step 3: At the lake bottom, P₁ = Pₐₜₘ + ρgh; at the surface, P₂ = Pₐₜₘ. Step 4: (Pₐₜₘ + ρgh)V₁ = 8PₐₜₘV₁ ⟹ ρgh = 7Pₐₜₘ. Step 5: The atmospheric pressure is Pₐₜₘ = ρgH, hence h = 7H. Final answer: 7H (Option C).