Question

A brass rod of length 50 cm and diameter 3.0 mm is joined to a steel rod of the same length and diameter. What is the change in length of the combined rod at 250 °C, if the original lengths are at 40.0 °C? Is there a ‘thermal stress’ developed at the junction? The ends of the rod are free to expand (Coefficient of linear expansion of brass = 2.0 × 10^–5 K–1, steel = 1.2 × 10^–5 K^–1 ).

Answer 1

Initial temperature, T₁ = 40°C
Final temperature, T₂ = 250°C
Change in temperature, ΔT = T₂ – T₁ = 210°C
Length of the brass rod at T₁, l₁ = 50 cm
Diameter of the brass rod at T₁, d₁ = 3.0 mm
Length of the steel rod at T₂, l₂ = 50 cm
Diameter of the steel rod at T₂, d₂ = 3.0 mm
Coefficient of linear expansion of brass, α₁ = 2.0 × 10^–5K^–1
Coefficient of linear expansion of steel, α2 = 1.2 × 10^–5K^–1
For the expansion in the brass rod, we have:
\(\frac{Change\, in \,length (Δ_{l1})}{Original\, length (l_1)}\) = α₁ΔT
∴ Δl₁ = 50 x (2.1 x 10^-5) x 210
= 0.2205 cm
For the expansion in the steel rod, we have:
Change in length (Δl2)/Original length (l₂) = α₂ΔT
∴ Δl₂ = 50 x (1.2 x 10-5) x 210
= 0.126 cm
Total change in the lengths of brass and steel,
Δl = Δl₁ + Δl₂ 
= 0.2205 + 0.126
= 0.346 cm
Total change in the length of the combined rod = 0.346 cm
Since the rod expands freely from both ends, no thermal stress is developed at the junction.