An Iron Ring on a Wheel

Q. A blacksmith fixes an iron ring on the rim of the wooden wheel of a horse cart. The diameter of the rim and the iron ring are 5.243 m and 5.231 m, respectively at 27 °C. Taking the coefficient of linear expansion of iron as 1.20 × 10⁻⁵ K⁻¹, to what temperature should the ring be heated so as to fit the rim of the wheel, in °C? expects 218 °C

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Q. A blacksmith fixes an iron ring on the rim of the wooden wheel of a horse cart. The diameter of the rim and the iron ring are 5.243 m $L_{2} = 5.243$ m and 5.231 m $L_{1} = 5.231$ m, respectively at 27 °C $T_{1} = 27$ °C. Taking the coefficient of linear expansion of iron as 1.20 × 10⁻⁵ K⁻¹ $α_{l} = 1.20 \times 10^{−5}$ K⁻¹, to what temperature should the ring be heated so as to fit the rim of the wheel, in °C?

The playground

Inputs

rim

L_{2}

5.24 m

ring

L_{1}

5.23 m

temperature

T_{1}

27 °C

expansion

α_{l}

1.2e-05 K⁻¹

Outputs

hot

T_{2}

The solution

Solved with ThermalExpansion

Given

$L_{1} = 5.231$ m
$L_{2} = 5.243$ m
$T_{1} = 27$ °C
$α_{l} = 1.20 \times 10^{−5}$ K⁻¹

Find $T_{2}$ .

Working

$T_{1} = 27$ °C $= 300.15$ K
linear expansion(ring) $L_{2} - L_{1} = α_{l} \cdot L_{1} \cdot (T_{2} - T_{1})$
$5.243 - 5.231 = 1.2 \times 10^{−5} \cdot 5.231 \cdot (T_{2} - 300.15)$
$T_{2} = 218.2$ °C
$T_{2} = 491.318037$ K $= 218.168037$ °C
$T_{2} \approx 218$ °Cto 3 significant figures

Conclusion

The ring has to grow by the small gap between it and the rim; iron grows by a few parts in a hundred thousand for each degree, so it takes about two hundred degrees.

The Ruby

1require_relative "../physics"23question :an_iron_ring_on_a_wheel do4  kind :numerical5  describes "A blacksmith fixes an iron ring on the rim of the wooden wheel of a horse cart. The diameter of the rim " \6            "and the iron ring are %{rim} and %{ring}, respectively at %{temperature}. Taking the coefficient of " \7            "linear expansion of iron as %{expansion}, %{hot}, in °C?"8  given rim: "5.243 m"9  given ring: "5.231 m"10  given temperature: "27 °C"11  given expansion: "1.20e-5 K-1"12  asks hot: "to what temperature should the ring be heated so as to fit the rim of the wheel"13end

1require_relative "../questions/an_iron_ring_on_a_wheel"2require_relative "../science/thermal_properties_of_matter/thermal_expansion"34solve :an_iron_ring_on_a_wheel, using: ThermalExpansion do5  objects :ring67  take ring: ring.length_at_first8  take rim: ring.length_after9  take temperature: ring.temperature_at_first10  take expansion: ring.coefficient_of_linear_expansion1112  hot = ring.temperature_after13  answer hot:, in: "°C", figures: 31415  conclusion "The ring has to grow by the small gap between it and the rim; iron grows by a few parts in " \16             "a hundred thousand for each degree, so it takes about two hundred degrees."17end

1require_relative "../solutions/an_iron_ring_on_a_wheel"23play_with :an_iron_ring_on_a_wheel do4  title "An Iron Ring on a Wheel"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 ring	$L_{2}$	m	—
length at first of ring	$L_{1}$	m	—
coefficient of linear expansion of ring	$α_{l}$	K⁻¹	—
temperature after of ring	$T_{2}$	K	—
temperature at first of ring	$T_{1}$	K	—
change in length of ground	$Δ l_{g}$	m	—
change in length of ring	$Δ l$	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(ring)

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(ring)