The coefficient of linear expansion of copper is 17 × 10^-⁶ K^-1and that of steel is 12 × 10^-6 K^-1. At 12°C a steel rod has a diameter of 2.540 cm and a copper pipe has a diameter of 2.536 cm. Which one of the following quantities is closest to the temperature to which the copper pipe must be heated in order for the unheated steel rod to fit snugly in the copper pipe? A) 53°C B) 81°C C) 93°C D) 105°C E) 143°C

Question

Quizplus · Accepted Answer

Let x be the temperature to which the copper pipe must be heated. Then the change in the diameter of the copper pipe is:
Δd = (diameter of copper pipe at 12°C) × αcopper × ΔT
where αcopper is the coefficient of linear expansion of copper and ΔT = x − 12 is the change in temperature.

Similarly, the change in the diameter of the steel rod is:
Δd = (diameter of steel rod at 12°C) × αsteel × ΔT
where αsteel is the coefficient of linear expansion of steel.

Since the steel rod has to fit snugly inside the copper pipe, we must have:
(diameter of steel rod at x°C) = (diameter of copper pipe at x°C)
Therefore, equating the two expressions for Δd and solving for x, we get:
(diameter of copper pipe at 12°C) × αcopper × ΔT = (diameter of steel rod at 12°C) × αsteel × ΔT
x = 12 + [(diameter of steel rod at 12°C) × αsteel]/[(diameter of copper pipe at 12°C) × αcopper]
Substituting the given values, we get:
x = 12 + [(2.540 cm) × (12 × 10^-6 K^-1)]/[(2.536 cm) × (17 × 10^-6 K^-1)]
x ≈ 105°C

Therefore, the temperature to which the copper pipe must be heated is closest to 105°C, which is answer choice D.

The Coefficient of Linear Expansion of Copper Is 17 ×