A Hole in a Copper Sheet

solved ThermalExpansion

Q. A hole is drilled in a copper sheet. The diameter of the hole is 4.24 cm $L_{1} = 4.24$ cm at 27.0 °C $T_{1} = 27.0$ °C. Coefficient of linear expansion of copper = 1.70 × 10⁻⁵ K⁻¹ $α_{l} = 1.70 \times 10^{−5}$ K⁻¹. What is the change in the diameter of the hole when the sheet is heated to 227 °C $T_{2} = 227$ °C, in cm?

Inputs

diameter

L_{1}

4.24 cm

cool

T_{1}

27 °C

expansion

α_{l}

1.7e-05 K⁻¹

hot

T_{2}

227 °C

Outputs

change

Δ l

How it's solved

Solved with ThermalExpansion

Given

$L_{1} = 4.24$ cm
$T_{1} = 27.0$ °C
$α_{l} = 1.70 \times 10^{−5}$ K⁻¹
$T_{2} = 227$ °C

Find $Δ l$ .

Working

$L_{1} = 4.24$ cm $= 0.0424$ m
$T_{1} = 27.0$ °C $= 300.15$ K
$T_{2} = 227$ °C $= 500.15$ K
linear expansion(hole) $L_{2} - L_{1} = α_{l} \cdot L_{1} \cdot (T_{2} - T_{1})$
$L_{2} - 0.0424 = 1.7 \times 10^{−5} \cdot 0.0424 \cdot (500.15 - 300.15)$
$L_{2} = 0.04254416$ m
the change is the length after less the length at first(hole) $Δ l = L_{2} - L_{1}$
$Δ l = 0.04254416 - 0.0424$
$Δ l = 0.01442$ cm
$Δ l = 1.4416 \times 10^{−4}$ m $= 0.014416$ cm
$Δ l \approx 0.0144$ cmto 3 significant figures

Conclusion

Heating swells the whole sheet like a photographic enlargement, so the hole grows too, just as a disc of copper filling it would.

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_hole_in_a_copper_sheet do4  kind :numerical5  describes "A hole is drilled in a copper sheet. The diameter of the hole is %{diameter} at %{cool}. Coefficient of " \6            "linear expansion of copper = %{expansion}. %{change} when the sheet is heated to %{hot}, in cm?"7  given diameter: "4.24 cm"8  given cool: "27.0 °C"9  given expansion: "1.70e-5 K-1"10  given hot: "227 °C"11  asks change: "What is the change in the diameter of the hole"12end

solution.rb

1require_relative "../questions/a_hole_in_a_copper_sheet"2require_relative "../science/thermal_properties_of_matter/thermal_expansion"34solve :a_hole_in_a_copper_sheet, using: ThermalExpansion do5  objects :hole67  take diameter: hole.length_at_first8  take cool: hole.temperature_at_first9  take expansion: hole.coefficient_of_linear_expansion10  take hot: hole.temperature_after1112  change = hole.change_in_length13  answer change:, in: "cm", figures: 31415  conclusion "Heating swells the whole sheet like a photographic enlargement, so the hole grows too, just " \16             "as a disc of copper filling it would."17end

playground.rb

1require_relative "../solutions/a_hole_in_a_copper_sheet"23play_with :a_hole_in_a_copper_sheet do4  title "A Hole in a Copper Sheet"5end

The laws it drew on

quantity	written	units	within	from
length at first	$L_{1}$	m	—	ThermalExpansion
length after	$L_{2}$	m	—	ThermalExpansion
change in length	$Δ l$	m	—	ThermalExpansion
temperature at first	$T_{1}$	K	—	ThermalExpansion
temperature after	$T_{2}$	K	—	ThermalExpansion
coefficient of linear expansion	$α_{l}$	K⁻¹	—	ThermalExpansion
length after of ground	$L_{2,g}$	m	—
length at first of ground	$L_{1,g}$	m	—
coefficient of linear expansion of ground	$α_{l,g}$	K⁻¹	—
temperature after of ground	$T_{2,g}$	K	—
temperature at first of ground	$T_{1,g}$	K	—
length after of hole	$L_{2}$	m	—
length at first of hole	$L_{1}$	m	—
coefficient of linear expansion of hole	$α_{l}$	K⁻¹	—
temperature after of hole	$T_{2}$	K	—
temperature at first of hole	$T_{1}$	K	—
change in length of ground	$Δ l_{g}$	m	—
change in length of hole	$Δ l$	m	—

L_{2,b} - L_{1,b} = α_{l,b} \cdot L_{1,b} \cdot (T_{2,b} - T_{1,b})

linear expansion

Δ l_{b} = L_{2,b} - L_{1,b}

the change is the length after less the length at first

L_{2,g} - L_{1,g} = α_{l,g} \cdot L_{1,g} \cdot (T_{2,g} - T_{1,g})

linear expansion(ground)

L_{2} - L_{1} = α_{l} \cdot L_{1} \cdot (T_{2} - T_{1})

linear expansion(hole)

Δ l_{g} = L_{2,g} - L_{1,g}

the change is the length after less the length at first(ground)

Δ l = L_{2} - L_{1}

the change is the length after less the length at first(hole)