A Copper Cube Under Pressure

solved BulkModulus

Q. Taking the bulk modulus of copper as 140 × 10⁹ N/m² $B = 140 \times 10^{9}$ N/m², what is the volume contraction of a solid copper cube, 10 cm $a = 10$ cm on an edge, when subjected to a hydraulic pressure of 7.0 × 10⁶ Pa $P_{g} = 7.0 \times 10^{6}$ Pa, in cm³?

Inputs

edge

a

10 cm

pressure

P_{g}

7e+06 Pa

modulus

B

1.4e+11 N/m²

Outputs

contraction

Δ V

How it's solved

Solved with BulkModulus

Given

$a = 10$ cm
$P_{g} = 7.0 \times 10^{6}$ Pa
$B = 140 \times 10^{9}$ N/m²

Find $Δ V$ .

Working

$a = 10$ cm $= 0.1$ m
$B = 140 \times 10^{9}$ N/m² $= 1.4 \times 10^{11}$ Pa
bulk modulus is pressure over volume strain(cube) $B = \frac{P_{g}}{Δ V_V}$
$1.4 \times 10^{11} = \frac{7 \times 10^{6}}{Δ V_V}$
$Δ V_V = 5 \times 10^{−5}$
a cube s volume(cube) $V = a^{3}$
$V = {0.1}^{3}$
$V = 0.001$ m³
fractional compression is the fall over the volume(cube) $Δ V_V = \frac{Δ V}{V}$
$5 \times 10^{−5} = \frac{Δ V}{0.001}$
$Δ V = 0.05$ cm³
$Δ V = 5 \times 10^{−8}$ m³ $= 0.05$ cm³
$Δ V \approx 0.050$ cm³to 2 significant figures

Conclusion

The cube loses the same fraction of its volume as the pressure is of copper's bulk modulus; copper is so stiff that the loss is a tiny sliver of the whole.

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 :a_copper_cube_under_pressure do4  kind :numerical5  describes "Taking the bulk modulus of copper as %{modulus}, %{contraction}, %{edge} on an edge, when subjected " \6            "to a hydraulic pressure of %{pressure}, in cm³?"7  given edge: "10 cm"8  given pressure: "7.0e6 Pa"9  given modulus: "140e9 N/m2"10  asks contraction: "what is the volume contraction of a solid copper cube"11end

solution.rb

1require_relative "../questions/a_copper_cube_under_pressure"2require_relative "../science/mechanical_properties_of_solids/bulk_modulus"34solve :a_copper_cube_under_pressure, using: BulkModulus do5  objects :cube67  take edge: cube.edge8  take pressure: cube.gauge_pressure9  take modulus: cube.bulk_modulus1011  contraction = cube.fall_in_volume12  answer contraction:, in: "cm3", figures: 21314  conclusion "The cube loses the same fraction of its volume as the pressure is of copper's bulk modulus; " \15             "copper is so stiff that the loss is a tiny sliver of the whole."16end

playground.rb

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

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(cube)

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

pressure adds the surface s(ground)

P = P_{a} + P_{g}

pressure adds the surface s(cube)

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(cube)

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

a cube s volume(ground)

V = a^{3}

a cube s volume(cube)

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(cube)

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(cube)