The Bottom of the Ocean

Q. The average depth of the Indian Ocean is about 3000 m. Given that the bulk modulus of water is 2.2 × 10⁹ N/m² and taking g as 10 m/s² and the density of water as 1000 kg/m³, what is the fractional compression, ΔV/V, of water at the bottom of the ocean? expects 0.0136

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Q. The average depth of the Indian Ocean is about 3000 m $h = 3000$ m. Given that the bulk modulus of water is 2.2 × 10⁹ N/m² $B = 2.2 \times 10^{9}$ N/m² and taking g as 10 m/s² $g = 10$ m/s² and the density of water as 1000 kg/m³ $ρ = 1000$ kg/m³, what is the fractional compression, ΔV/V, of water at the bottom of the ocean?

The playground

Inputs

depth

h

3000.00 m

modulus

B

2.2e+09 N/m²

gravity

g

10 m/s²

density

ρ

1000 kg/m³

Outputs

compression

Δ V_V

The solution

Solved with BulkModulus

Given

$h = 3000$ m
$B = 2.2 \times 10^{9}$ N/m²
$g = 10$ m/s²
$ρ = 1000$ kg/m³

Find $Δ V_V$ .

Working

$B = 2.2 \times 10^{9}$ N/m² $= 2.2 \times 10^{9}$ Pa
pressure of the fluid above(bottom water) $P_{g} = ρ \cdot g \cdot h$
$P_{g} = 1000 \cdot 10 \cdot 3000$
$P_{g} = 3 \times 10^{7}$ Pa
bulk modulus is pressure over volume strain(bottom water) $B = \frac{P_{g}}{Δ V_V}$
$2.2 \times 10^{9} = \frac{3 \times 10^{7}}{Δ V_V}$
$Δ V_V = 0.0136363636$
$Δ V_V \approx 0.0136$ to 3 significant figures

Conclusion

Three kilometres of water press hard, but water is hard to squeeze: the pressure over the bulk modulus is barely more than a percent.

The Ruby

1require_relative "../physics"23question :the_bottom_of_the_ocean do4  kind :numerical5  describes "The average depth of the Indian Ocean is about %{depth}. Given that the bulk modulus of water is " \6            "%{modulus} and taking g as %{gravity} and the density of water as %{density}, %{compression}?"7  given depth: "3000 m"8  given modulus: "2.2e9 N/m2"9  given gravity: "10 m/s2"10  given density: "1000 kg/m3"11  asks compression: "what is the fractional compression, ΔV/V, of water at the bottom of the ocean"12end

1require_relative "../questions/the_bottom_of_the_ocean"2require_relative "../science/mechanical_properties_of_solids/bulk_modulus"34solve :the_bottom_of_the_ocean, using: BulkModulus do5  objects :bottom_water67  take depth: bottom_water.depth8  take modulus: bottom_water.bulk_modulus9  take gravity: g10  take density: rho1112  compression = bottom_water.fractional_compression13  answer compression:, figures: 31415  conclusion "Three kilometres of water press hard, but water is hard to squeeze: the pressure over the " \16             "bulk modulus is barely more than a percent."17end

1require_relative "../solutions/the_bottom_of_the_ocean"23play_with :the_bottom_of_the_ocean do4  title "The Bottom of the Ocean"5end

The laws

quantity	written	units	within	from
mass	$m$	kg	0 … ∞	Pressure
gravity	$g$	m/s²	—	Pressure
thrust	$F$	N	—	Pressure
area	$A$	m²	—	Pressure
pressure	$P$	Pa	0 … ∞	Pressure
pressure at the surface	$P_{a}$	Pa	—	PressureAtDepth
density	$ρ$	kg/m³	0 … ∞	PressureAtDepth
depth	$h$	m	—	PressureAtDepth
gauge pressure	$P_{g}$	Pa	—	PressureAtDepth
bulk modulus	$B$	Pa	—	BulkModulus
fractional compression	$Δ V_V$	—	—	BulkModulus
edge	$a$	m	—	BulkModulus
volume	$V$	m³	0 … ∞	BulkModulus
fall in volume	$Δ V$	m³	—	BulkModulus
density when squeezed	$ρ_{B}$	kg/m³	—	BulkModulus
pressure of ground	$P_{g}$	Pa	0 … ∞
thrust on ground	$F_{g}$	N	—
area of ground	$A_{g}$	m²	—
pressure of bottom water	$P$	Pa	0 … ∞
thrust on bottom water	$F$	N	—
area of bottom water	$A$	m²	—
gauge pressure on ground	$P_{g,g}$	Pa	—
depth of ground	$h_{g}$	m	—
gauge pressure on bottom water	$P_{g}$	Pa	—
depth of bottom water	$h$	m	—
bulk modulus of ground	$B_{g}$	Pa	—
fractional compression of ground	$Δ {V_V}_{g}$	—	—
bulk modulus of bottom water	$B$	Pa	—
fractional compression of bottom water	$Δ V_V$	—	—
volume of ground	$V_{g}$	m³	0 … ∞
edge of ground	$a_{g}$	m	—
volume of bottom water	$V$	m³	0 … ∞
edge of bottom water	$a$	m	—
fall in volume of ground	$Δ V_{g}$	m³	—
fall in volume of bottom water	$Δ V$	m³	—
density when squeezed of ground	$ρ_{B,g}$	kg/m³	—
density when squeezed of bottom water	$ρ_{B}$	kg/m³	—
mass of bottom water	$m$	kg	0 … ∞

P_{s} = \frac{F_{s}}{A_{s}}

pressure is force over area

P_{g,p} = ρ \cdot g \cdot h_{p}

pressure of the fluid above

P_{p} = P_{a} + P_{g,p}

pressure adds the surface s

B_{b} = \frac{P_{g,b}}{Δ {V_V}_{b}}

bulk modulus is pressure over volume strain

V_{b} = {a_{b}}^{3}

a cube s volume

Δ {V_V}_{b} = \frac{Δ V_{b}}{V_{b}}

fractional compression is the fall over the volume

ρ_{B,b} = \frac{ρ}{1 - Δ {V_V}_{b}}

the same mass in less room

P_{g} = \frac{F_{g}}{A_{g}}

pressure is force over area(ground)

P = \frac{F}{A}

pressure is force over area(bottom water)

P_{g,g} = ρ \cdot g \cdot h_{g}

pressure of the fluid above(ground)

P_{g} = ρ \cdot g \cdot h

pressure of the fluid above(bottom water)

P_{g} = P_{a} + P_{g,g}

pressure adds the surface s(ground)

P = P_{a} + P_{g}

pressure adds the surface s(bottom water)

B_{g} = \frac{P_{g,g}}{Δ {V_V}_{g}}

bulk modulus is pressure over volume strain(ground)

B = \frac{P_{g}}{Δ V_V}

bulk modulus is pressure over volume strain(bottom water)

V_{g} = {a_{g}}^{3}

a cube s volume(ground)

V = a^{3}

a cube s volume(bottom water)

Δ {V_V}_{g} = \frac{Δ V_{g}}{V_{g}}

fractional compression is the fall over the volume(ground)

Δ V_V = \frac{Δ V}{V}

fractional compression is the fall over the volume(bottom water)

ρ_{B,g} = \frac{ρ}{1 - Δ {V_V}_{g}}

the same mass in less room(ground)

ρ_{B} = \frac{ρ}{1 - Δ V_V}

the same mass in less room(bottom water)