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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).
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Solution 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:
`("Change in length"(trianglel_1))/("Original length"(l_1))" = alpha_1triangleT`
`:. trianglel_1 = 50 xx (2.1xx 10^(-5))xx210`
= 0.2205 cm
For the expansion in the steel rod, we have:
`("Change in length"(trianglel_2))/("Original length"(l_2))" = alpha_2triangleT`
`:.trianglel_2 = 50xx (1.2xx10^(-5))xx210`
= 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.
Solution 2
Here `l_"brass" = l_"steel" = 50 cm, d_"brass" = d_"steel" = 3mm = 0.3 cm, trianglel_"brass" = ?, trianglel_"steel" = ?`
`triangleT = 250 - 40 = 210 ""^@C`
`alpha_"brass" = 2xx10^(-5) ""^@C^(-1) and alpha_"steel" = 1.2 xx 10^(-5) ""^@C^(-1)`
Now `trianglel_"brass" = alpha_"brass" xx l_"brass" xx triangleT`
`= 2xx10^(-5) xx 50 xx 210 = 0.21 cm`
Now `trianglel_"steel" = alpha_"steel" xx l_"steel" xx triangleT`
`= 1.2 xx 10^(-5) xx 50 xx 210`
`=0.126 cm = 0.13 cm`
`:. "Total change in length," trianglel = trianglel_"brass" + trianglel_"steel" = 0.21 + 0.13 = 0.34 cm`
Since the rod is not clamped at its ends, no thermal stress developed at the junction.
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