At a certain temperature, a rod is hung from an aluminum frame, as the drawing shows. A small gap of 1 mm exists between the rod and the floor. The frame and rod are heated uniformly. At what temperat

Question

At a certain temperature, a rod is hung from an aluminum frame, as the drawing shows. A small gap of 1 mm exists between the rod and the floor. The frame and rod are heated uniformly. At what temperature change (in ∘ C) will the rod touch the floor, assuming that the rod is made from lead? The coefficient of thermal expansion of aluminum is 23 × 1 0 − 6 / ∘ C and for lead it is 29 × 1 0 − 6 / ∘ C. The height of the frame is 10 cm.

TestHub · Accepted Answer

Frame height H = 10 cm; rod length L = 10 − 0.1 = 9.9 cm (gap is 1 mm). Gap closes when rod growth minus frame growth equals the gap: L α P b ​ Δ T − H α A l ​ Δ T = 0.1 cm Δ T [ 9.9 ( 29 × 1 0 − 6 ) − 10 ( 23 × 1 0 − 6 )] = 0.1 Δ T ( 287.1 − 230 ) × 1 0 − 6 = 0.1 ⇒ Δ T = 57.1 × 1 0 − 6 0.1 ​ ≈ 1751. 3 ∘ C Answer: about 1751 to 1752 (accepted range 1751.00 to 1752.00).

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