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Question

What should be lengths of steel and copper rod, so that the length of steel rod is 5 cm longer than the copper rod at all temperatures? (α of copper 1.7×105oC1 and α for steel =1.1×105oC1).

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Solution

Let α1 and α2 be the coefficients of linear expansion of steel and copper respectively. Suppose that the steel rod is of length l1 and cooper rod of length l2 so that all temperatures, the steel rod is 5 cm loner than the copper rod i.e.
l1l2=5cm.....(1)
It will happen so, if on increase of temperature say by θoC, the increase in length of the two rods is same i.e.
Δl1=Δl2
l1α1θ=l2α2θ
l2=α1α2l1
Here, α1=1.1×105oC1,
α2=1.7×105C1
l2=1.1×1051.7×105l1=1117l1
In the equation, (1) substituting for l2, we have
l11117l1=5
l1=856=14.17cm
Also, l2=l15=14.175=9.17cm.

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