A Hole in a Copper Sheet

Q. A hole is drilled in a copper sheet. The diameter of the hole is 4.24 cm at 27.0 °C. Coefficient of linear expansion of copper = 1.70 × 10⁻⁵ K⁻¹. What is the change in the diameter of the hole when the sheet is heated to 227 °C, in cm? expects 0.0144 cm

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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?

The playground

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

The solution

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 Ruby

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

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

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

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)