A uniform steel rod has length ℓ at 0 ∘ C . Now one of its end is kept in ice ( 0 ∘ C ) and the other end is kept in steam ( 10 0 ∘ C ) . If the coefficient of thermal expansion of the rod is α , how

Question

A uniform steel rod has length ℓ at 0 ∘ C . Now one of its end is kept in ice ( 0 ∘ C ) and the other end is kept in steam ( 10 0 ∘ C ) . If the coefficient of thermal expansion of the rod is α , how much is the thermal expansion of the rod at steady state?

TestHub · Accepted Answer

T ( x ) = 0 + ℓ 100 − 0 ​ x Let us take a small element of length ' dx '. Expansion in the small element ​ d ( e ) = ( d x ) α ( T ( x ) − 0 ) ​ d ( e ) = ( d x ) α ​ e = ℓ 100 α ​ ∫ x = 0 x = ℓ ​ x d x ​ e = 50 α ℓ . ​ Alternate solution : As the expansion has linear relation with increase in temperature, so we can use average ​ e = ℓ ∝ ( Δ T ) ​ e = ℓ ∝ ( 2 0 + 100 ​ ) ​ e = 50 ℓ α . ​

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