The Bottom of the Ocean

solved BulkModulus

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?

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

How it's solved

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 web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

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

solution.rb

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

playground.rb

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 it drew on

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)