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 (Co-efficient 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:

For the expansion in the steel rod, we have:

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.