A Human Pyramid — phryzby

solved YoungsModulus

Q. In a human pyramid in a circus, the entire weight of the balanced group is supported by the legs $F_{t} = \frac{(m_{g} - m_{p}) \cdot g}{2}$ of a performer who is lying on his back. The combined mass of all the persons performing the act, and the tables, plaques etc. involved is 280 kg $m_{g} = 280$ kg. The mass of the performer lying on his back at the bottom of the pyramid is 60 kg $m_{p} = 60$ kg. Each thighbone (femur) of this performer has a length of 50 cm $L_{t} = 50$ cm and an effective radius of 2.0 cm $r_{t} = 2.0$ cm. Taking Young's modulus of bone as 9.4 × 10⁹ N/m² $Y_{t} = 9.4 \times 10^{9}$ N/m² and g as 9.8 m/s² $g = 9.8$ m/s², by how much does each thighbone get compressed under the extra load, in cm?

Inputs

group mass

m_{g}

280 kg

performer mass

m_{p}

60 kg

length

L_{t}

50 cm

radius

r_{t}

2 cm

modulus

Y_{t}

9.4e+09 N/m²

gravity

g

9.8 m/s²

Outputs

compression

Δ L_{t}

How it's solved

Solved with YoungsModulus

Given

$m_{g} = 280$ kg
$m_{p} = 60$ kg
$L_{t} = 50$ cm
$r_{t} = 2.0$ cm
$Y_{t} = 9.4 \times 10^{9}$ N/m²
$g = 9.8$ m/s²
$F_{t} = \frac{(m_{g} - m_{p}) \cdot g}{2}$ from “supported by the legs”

Find $Δ L_{t}$ .

Working

$L_{t} = 50$ cm $= 0.5$ m
$r_{t} = 2.0$ cm $= 0.02$ m
$Y_{t} = 9.4 \times 10^{9}$ N/m² $= 9.4 \times 10^{9}$ Pa
supported by the legs $F_{t} = \frac{(m_{g} - m_{p}) \cdot g}{2}$
$F_{t} = \frac{(280 - 60) \cdot 9.8}{2}$
$F_{t} = 1078$ N
a round rod s cross section(thighbone) $A_{t} = π \cdot {r_{t}}^{2}$
$A_{t} = 3.14159265 \cdot {0.02}^{2}$
$A_{t} = 0.00125663706$ m²
stress is load over area(thighbone) $σ_{t} = \frac{F_{t}}{A_{t}}$
$σ_{t} = \frac{1078}{0.00125663706}$
$σ_{t} = 857845.143$ Pa
hookes law(thighbone) $Y_{t} = \frac{σ_{t}}{ε_{t}}$
$9.4 \times 10^{9} = \frac{857845.143}{ε_{t}}$
$ε_{t} = 9.12601216 \times 10^{−5}$
strain is stretch over length(thighbone) $ε_{t} = \frac{Δ L_{t}}{L_{t}}$
$9.12601216 \times 10^{−5} = \frac{Δ L_{t}}{0.5}$
$Δ L_{t} = 0.004563$ cm
$Δ L_{t} = 4.56300608 \times 10^{−5}$ m $= 0.00456300608$ cm
$Δ L_{t} \approx 0.00456$ cmto 3 significant figures

Conclusion

Each thighbone carries half of everyone else's weight, and bone is stiff enough that it shortens by a tiny fraction of its length.

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_human_pyramid do4  kind :numerical5  describes "In a human pyramid in a circus, the entire weight of the balanced group is supported by the legs of a " \6            "performer who is lying on his back. The combined mass of all the persons performing the act, and the " \7            "tables, plaques etc. involved is %{group_mass}. The mass of the performer lying on his back at the " \8            "bottom of the pyramid is %{performer_mass}. Each thighbone (femur) of this performer has a length of " \9            "%{length} and an effective radius of %{radius}. Taking Young's modulus of bone as %{modulus} and g as " \10            "%{gravity}, %{compression}, in cm?"11  given group_mass: "280 kg"12  given performer_mass: "60 kg"13  given length: "50 cm"14  given radius: "2.0 cm"15  given modulus: "9.4e9 N/m2"16  given gravity: "9.8 m/s2"17  asks compression: "by how much does each thighbone get compressed under the extra load"18end

solution.rb

1require_relative "../questions/a_human_pyramid"2require_relative "../science/mechanical_properties_of_solids/youngs_modulus"34solve :a_human_pyramid, using: YoungsModulus do5  objects :group, :performer, :thighbone67  take group_mass: group.mass8  take performer_mass: performer.mass9  take length: thighbone.original_length10  take radius: thighbone.thickness11  take modulus: thighbone.youngs_modulus12  take gravity: g1314  deduce_from "supported by the legs", thighbone.load => (group.mass - performer.mass) * g / 21516  compression = thighbone.extension17  answer compression:, in: "cm", figures: 31819  conclusion "Each thighbone carries half of everyone else's weight, and bone is stiff enough that it " \20             "shortens by a tiny fraction of its length."21end

playground.rb

1require_relative "../solutions/a_human_pyramid"23play_with :a_human_pyramid do4  title "A Human Pyramid"5end

The laws it drew on

quantity	written	units	within	from
pi	$π$	—	0 … ∞	YoungsModulus
mass	$m$	kg	0 … ∞	YoungsModulus
gravity	$g$	m/s²	—	YoungsModulus
load	$F$	N	—	YoungsModulus
thickness	$r$	m	0 … ∞	YoungsModulus
diameter	$D$	m	—	YoungsModulus
cross section	$A$	m²	—	YoungsModulus
original length	$L$	m	—	YoungsModulus
extension	$Δ L$	m	—	YoungsModulus
stress	$σ$	Pa	—	YoungsModulus
strain	$ε$	—	—	YoungsModulus
youngs modulus	$Y$	Pa	—	YoungsModulus
thickness of ground	$r_{g}$	m	0 … ∞
diameter of ground	$D_{g}$	m	—
thickness of group	$r_{g}$	m	0 … ∞
diameter of group	$D_{g}$	m	—
thickness of performer	$r_{p}$	m	0 … ∞
diameter of performer	$D_{p}$	m	—
thickness of thighbone	$r_{t}$	m	0 … ∞
diameter of thighbone	$D_{t}$	m	—
cross section of ground	$A_{g}$	m²	—
cross section of group	$A_{g}$	m²	—
cross section of performer	$A_{p}$	m²	—
cross section of thighbone	$A_{t}$	m²	—
stress of ground	$σ_{g}$	Pa	—
load on ground	$F_{g}$	N	—
stress of group	$σ_{g}$	Pa	—
load on group	$F_{g}$	N	—
stress of performer	$σ_{p}$	Pa	—
load on performer	$F_{p}$	N	—
stress of thighbone	$σ_{t}$	Pa	—
load on thighbone	$F_{t}$	N	—
strain of ground	$ε_{g}$	—	—
extension of ground	$Δ L_{g}$	m	—
original length of ground	$L_{g}$	m	—
strain of group	$ε_{g}$	—	—
extension of group	$Δ L_{g}$	m	—
original length of group	$L_{g}$	m	—
strain of performer	$ε_{p}$	—	—
extension of performer	$Δ L_{p}$	m	—
original length of performer	$L_{p}$	m	—
strain of thighbone	$ε_{t}$	—	—
extension of thighbone	$Δ L_{t}$	m	—
original length of thighbone	$L_{t}$	m	—
youngs modulus of ground	$Y_{g}$	Pa	—
youngs modulus of group	$Y_{g}$	Pa	—
youngs modulus of performer	$Y_{p}$	Pa	—
youngs modulus of thighbone	$Y_{t}$	Pa	—
mass of group	$m_{g}$	kg	0 … ∞
mass of performer	$m_{p}$	kg	0 … ∞
mass of thighbone	$m_{t}$	kg	0 … ∞

r_{r} = \frac{D_{r}}{2}

radius is half the diameter

A_{r} = π \cdot {r_{r}}^{2}

a round rod s cross section

σ_{r} = \frac{F_{r}}{A_{r}}

stress is load over area

ε_{r} = \frac{Δ L_{r}}{L_{r}}

strain is stretch over length

Y_{r} = \frac{σ_{r}}{ε_{r}}

hookes law

r_{g} = \frac{D_{g}}{2}

radius is half the diameter(ground)

r_{g} = \frac{D_{g}}{2}

radius is half the diameter(group)

r_{p} = \frac{D_{p}}{2}

radius is half the diameter(performer)

r_{t} = \frac{D_{t}}{2}

radius is half the diameter(thighbone)

A_{g} = π \cdot {r_{g}}^{2}

a round rod s cross section(ground)

A_{g} = π \cdot {r_{g}}^{2}

a round rod s cross section(group)

A_{p} = π \cdot {r_{p}}^{2}

a round rod s cross section(performer)

A_{t} = π \cdot {r_{t}}^{2}

a round rod s cross section(thighbone)

σ_{g} = \frac{F_{g}}{A_{g}}

stress is load over area(ground)

σ_{g} = \frac{F_{g}}{A_{g}}

stress is load over area(group)

σ_{p} = \frac{F_{p}}{A_{p}}

stress is load over area(performer)

σ_{t} = \frac{F_{t}}{A_{t}}

stress is load over area(thighbone)

ε_{g} = \frac{Δ L_{g}}{L_{g}}

strain is stretch over length(ground)

ε_{g} = \frac{Δ L_{g}}{L_{g}}

strain is stretch over length(group)

ε_{p} = \frac{Δ L_{p}}{L_{p}}

strain is stretch over length(performer)

ε_{t} = \frac{Δ L_{t}}{L_{t}}

strain is stretch over length(thighbone)

Y_{g} = \frac{σ_{g}}{ε_{g}}

hookes law(ground)

Y_{g} = \frac{σ_{g}}{ε_{g}}

hookes law(group)

Y_{p} = \frac{σ_{p}}{ε_{p}}

hookes law(performer)

Y_{t} = \frac{σ_{t}}{ε_{t}}

hookes law(thighbone)

F_{t} = \frac{(m_{g} - m_{p}) \cdot g}{2}

supported by the legs