A Glass Slab Under Pressure

Q. Taking the bulk modulus of glass as 37 × 10⁹ N/m², what is the fractional change in volume of a glass slab, when subjected to a hydraulic pressure of 10 atm? expects 2.74 × 10⁻⁵

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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?

The playground

Inputs

pressure

P_{g}

10 atm

modulus

B

3.7e+10 N/m²

Outputs

compression

Δ V_V

The solution

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 Ruby

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

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

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

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)