If average depth of an ocean is 4000 m and the bulk modulus of water is 2 × 10⁹ Nm⁻², then fractional compression (Δ V)/V of water at the bottom of ocean is α × 10⁻². The value of α is _______ (Given, g=10 ms⁻², ρ=1000 kg m⁻³)

Solution

Physics sir · 3 steps

Water at the bottom feels pressure ρ g h; divide it by the bulk modulus to get the squeeze.

1. Pressure at bottom. The extra pressure on water at depth h is P = ρ g h = 1000 × 10 × 4000 = 4 × 10⁷ N m⁻².
2. Use bulk modulus. Bulk modulus is pressure divided by fractional volume change, B = P/(Δ V / V). Rearrange it to get the compression.

   (Δ V)/V = P/B = (4 × 10⁷)/(2 × 10⁹) = 2 × 10⁻²

3. So. Matching α × 10⁻² with 2 × 10⁻² gives α = 2.

The common slip: 0.02. Reporting the whole fraction (Δ V)/V gives 0.02, but the question asks only for α, the number in front of 10⁻².

The idea to keep. Bulk modulus links pressure to fractional volume change: (Δ V)/V = P/B. For a fluid at depth, the pressure to use is ρ g h.

Answer from the NTA final answer key. Solution by PrepWiser.