The given lengths at 0 °C are 2.5 m
Let l₀ be the given lengths of the glass and steel rods at 0 °C. Let l and l' be the lengths of the glass and steel rods at 100 °C respectively.
From our expression for linear expansivity,
l = l₀ + l₀αΔθ where α = linear expansivity of glass = 0.000008/°C and Δθ = temperature change = θ - θ' where θ = 100 °C and θ' = 0 °C. So, Δθ = 100 °C - 0 °C = 100 °C.
Also,
l' = l₀ + l₀α'Δθ where α' = linear expansivity of steel = 0.000012/°C and Δθ = temperature change = θ - θ' where θ = 100 °C and θ' = 0 °C. So, Δθ = 100 °C - 0 °C = 100 °C.
Since the difference in their lengths at 100 °C = 0.001 m, we have that
l - l' = l₀ + l₀αΔθ - (l₀ + l₀α'Δθ)
l - l' = l₀ + l₀αΔθ - l₀ - l₀α'Δθ)
l - l' = l₀αΔθ - l₀α'Δθ
l - l' = l₀(α- α')Δθ
Making l₀ subject of the formula, we have
l₀ = (l - l')/[(α- α')Δθ]
Substituting the values of the variables into the equation, we have
l₀ = (l - l')/[(α- α')Δθ]
l₀ = 0.001 m/[(0.000008/°C - 0.000012/°C)100 °C.]
l₀ = 0.001 m/[(-0.000004/°C)100 °C.]
l₀ = 0.001 m/-0.0004
l₀ = -2.5 m
Neglecting the negative sign,
l₀ = 2.5 m
So, the given lengths at 0 °C are 2.5 m
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