A Glass Slab Under Pressure

solved BulkModulus

Q. Taking the bulk modulus of glass as 37 × 10⁹ N/m² $B = 37 \times 10^{9}$ N/m², what is the fractional change in volume of a glass slab, when subjected to a hydraulic pressure of 10 atm $P_{g} = 10$ atm?

Inputs

pressure

P_{g}

10 atm

modulus

B

3.7e+10 N/m²

Outputs

compression

Δ V_V

How it's solved

Solved with BulkModulus

Given

$P_{g} = 10$ atm
$B = 37 \times 10^{9}$ N/m²

Find $Δ V_V$ .

Working

$P_{g} = 10$ atm $= 1.01325 \times 10^{6}$ Pa
$B = 37 \times 10^{9}$ N/m² $= 3.7 \times 10^{10}$ Pa
bulk modulus is pressure over volume strain(slab) $B = \frac{P_{g}}{Δ V_V}$
$3.7 \times 10^{10} = \frac{1.01325 \times 10^{6}}{Δ V_V}$
$Δ V_V = 2.73851351 \times 10^{−5}$
$Δ V_V \approx 2.74 \times 10^{−5}$ to 3 significant figures

Conclusion

The fractional change in volume is the pressure over the bulk modulus; glass is so hard to squeeze that ten atmospheres barely change it.

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_glass_slab_under_pressure do4  kind :numerical5  describes "Taking the bulk modulus of glass as %{modulus}, %{compression}, when subjected to a hydraulic " \6            "pressure of %{pressure}?"7  given pressure: "10 atm"8  given modulus: "37e9 N/m2"9  asks compression: "what is the fractional change in volume of a glass slab"10end

solution.rb

1require_relative "../questions/a_glass_slab_under_pressure"2require_relative "../science/mechanical_properties_of_solids/bulk_modulus"34solve :a_glass_slab_under_pressure, using: BulkModulus do5  objects :slab67  take pressure: slab.gauge_pressure8  take modulus: slab.bulk_modulus910  compression = slab.fractional_compression11  answer compression:, figures: 31213  conclusion "The fractional change in volume is the pressure over the bulk modulus; glass is so hard to " \14             "squeeze that ten atmospheres barely change it."15end

playground.rb

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

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(slab)

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

pressure adds the surface s(ground)

P = P_{a} + P_{g}

pressure adds the surface s(slab)

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(slab)

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

a cube s volume(ground)

V = a^{3}

a cube s volume(slab)

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(slab)

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(slab)