A brass rod of length 50 cm and diameter 3.0 mm is joined to a steel r

1.	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 x 10-5°C-1, steel = 1.2 x 10-5°C-1).
Answer» Here, l_brass = l_steel = 50 cm d_brass = d_steel = 3 mm = 0.3 cm Δl_brass = ? Δl_steel = ? ΔT = 250 - 40 = 210°C α_brass = 2 x 10^-5°C^-1 and α_steel = 1.2 x 10^-5°C^-1 Now, Δl_brass = α_brass x l_brass x ΔT = 2 x 10^-5 x 50 x 210 = 0.21 cm Again, Δl_steel = α_steel x l_steel x ΔT = 1.2 x 10^-5 x 50 x 210 = 0.126 cm ≈ 0.13 cm ∴ Total change in length, Δl = Δl_brass + Δl_steel = 0.21 + 0.13 = 0.34 cm Since, the rod is not clamped at its ends no thermal stress is developed at the junction.

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 x 10-5°C-1, steel = 1.2 x 10-5°C-1).

Answer»

Here, l_brass = l_steel = 50 cm

d_brass = d_steel = 3 mm = 0.3 cm

Δl_brass = ?

Δl_steel = ?

ΔT = 250 - 40 = 210°C

α_brass = 2 x 10^-5°C^-1

and α_steel = 1.2 x 10^-5°C^-1

Now, Δl_brass = α_brass x l_brass x ΔT

= 2 x 10^-5 x 50 x 210 = 0.21 cm

Again, Δl_steel = α_steel x l_steel x ΔT

= 1.2 x 10^-5 x 50 x 210 = 0.126 cm

≈ 0.13 cm

∴ Total change in length, Δl = Δl_brass + Δl_steel

= 0.21 + 0.13 = 0.34 cm

Since, the rod is not clamped at its ends no thermal stress is developed at the junction.