__Question 11.10__:
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}).

__Solution__:*T*

_{1}= 40°C

Final temperature,

*T*

_{2}= 250°C

Change in temperature, Δ

*T*=

*T*

_{2}–

*T*

_{1 }= 210°C

Length of the brass rod at

*T*

_{1},

*l*

_{1}= 50 cm

Diameter of the brass rod at

*T*

_{1},

*d*

_{1}= 3.0 mm

Length of the steel rod at

*T*

_{2},

*l*

_{2}= 50 cm

Diameter of the steel rod at

*T*

_{2},

*d*

_{2}= 3.0 mm

Coefficient of linear expansion of brass, α

_{1}= 2.0 × 10

^{–5}K

^{–1}

Coefficient of linear expansion of steel, α

_{2}= 1.2 × 10

^{–5}K

^{–1}

For the expansion in the brass rod, we have:

Change in length (∆l

_{1}) / Original length (l

_{1}) = α

_{1}Δ

*T*

∴ ∆l

_{1}= 50 X (2.1 X 10

^{-5}) X 210

= 0.2205 cm

For the expansion in the steel rod, we have:

Change in length (∆l

_{2}) / Original length (l

_{2}) = α

_{2}Δ

*T*

∴ ∆l

_{1}= 50 X (1.2 X 10

^{-5}) X 210

= 0.126 cm

Total change in the lengths of brass and steel,

Δ

*l*= Δ

*l*

_{1}+ Δ

*l*

_{2 }

= 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.

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