Water Squeezed at Depth

solved BulkModulus

Q. Taking the bulk modulus of water as 2.2 × 10⁹ N/m² $B = 2.2 \times 10^{9}$ N/m², what is the density of water at a depth where pressure is 80.0 atm $P_{g} = 80.0$ atm, given that its density at the surface is 1.03 × 10³ kg/m³ $ρ = 1.03 \times 10^{3}$ kg/m³, in kg/m³?

Inputs

pressure

P_{g}

80 atm

density

ρ

1030 kg/m³

modulus

B

2.2e+09 N/m²

Outputs

squeezed

ρ_{B}

How it's solved

Solved with BulkModulus

Given

$P_{g} = 80.0$ atm
$ρ = 1.03 \times 10^{3}$ kg/m³
$B = 2.2 \times 10^{9}$ N/m²

Find $ρ_{B}$ .

Working

$P_{g} = 80.0$ atm $= 8.106 \times 10^{6}$ Pa
$B = 2.2 \times 10^{9}$ N/m² $= 2.2 \times 10^{9}$ Pa
bulk modulus is pressure over volume strain(deep water) $B = \frac{P_{g}}{Δ V_V}$
$2.2 \times 10^{9} = \frac{8.106 \times 10^{6}}{Δ V_V}$
$Δ V_V = 0.00368454545$
the same mass in less room(deep water) $ρ_{B} = \frac{ρ}{1 - Δ V_V}$
$ρ_{B} = \frac{1030}{1 - 0.00368454545}$
$ρ_{B} = 1034$ kg/m³
$ρ_{B} \approx 1034$ kg/m³to 4 significant figures

Conclusion

The pressure squeezes the water into a slightly smaller volume, and the same mass in less room is slightly denser; water is so stiff that the change is a fraction of 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 :water_squeezed_at_depth do4  kind :numerical5  describes "Taking the bulk modulus of water as %{modulus}, %{squeezed} at a depth where pressure is %{pressure}, " \6            "given that its density at the surface is %{density}, in kg/m³?"7  given pressure: "80.0 atm"8  given density: "1.03e3 kg/m3"9  given modulus: "2.2e9 N/m2"10  asks squeezed: "what is the density of water"11end

solution.rb

1require_relative "../questions/water_squeezed_at_depth"2require_relative "../science/mechanical_properties_of_solids/bulk_modulus"34solve :water_squeezed_at_depth, using: BulkModulus do5  objects :deep_water67  take pressure: deep_water.gauge_pressure8  take density: rho9  take modulus: deep_water.bulk_modulus1011  squeezed = deep_water.density_when_squeezed12  answer squeezed:, in: "kg/m3", figures: 41314  conclusion "The pressure squeezes the water into a slightly smaller volume, and the same mass in less " \15             "room is slightly denser; water is so stiff that the change is a fraction of a percent."16end

playground.rb

1require_relative "../solutions/water_squeezed_at_depth"23play_with :water_squeezed_at_depth do4  title "Water Squeezed at Depth"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 deep water	$P$	Pa	0 … ∞
thrust on deep water	$F$	N	—
area of deep water	$A$	m²	—
gauge pressure on ground	$P_{g,g}$	Pa	—
depth of ground	$h_{g}$	m	—
gauge pressure on deep water	$P_{g}$	Pa	—
depth of deep water	$h$	m	—
bulk modulus of ground	$B_{g}$	Pa	—
fractional compression of ground	$Δ {V_V}_{g}$	—	—
bulk modulus of deep water	$B$	Pa	—
fractional compression of deep water	$Δ V_V$	—	—
volume of ground	$V_{g}$	m³	0 … ∞
edge of ground	$a_{g}$	m	—
volume of deep water	$V$	m³	0 … ∞
edge of deep water	$a$	m	—
fall in volume of ground	$Δ V_{g}$	m³	—
fall in volume of deep water	$Δ V$	m³	—
density when squeezed of ground	$ρ_{B,g}$	kg/m³	—
density when squeezed of deep water	$ρ_{B}$	kg/m³	—
mass of deep 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(deep 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(deep water)

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

pressure adds the surface s(ground)

P = P_{a} + P_{g}

pressure adds the surface s(deep 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(deep water)

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

a cube s volume(ground)

V = a^{3}

a cube s volume(deep 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(deep 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(deep water)