A hole is drilled in a copper sheet. The diameter of the hole is 4.24 cm at 27.0 °C. What is the change in the diameter of the hole when the sheet is heated to 227 °C? Coefficient of linear expansion of copper = .
Given: Initial diameter of hole, D₁ = 4.24 cm; Initial temperature, T₁ = 27.0 °C; Final temperature, T₂ = 227 °C; Coefficient of linear expansion of copper, α = ⁻⁵ K⁻¹.
Change in temperature: ΔT = T₂ - T₁ = 227 - 27 = 200 °C = 200 K
Formula: When a sheet with a hole is heated, the hole expands as if it were made of the same material. Using linear expansion formula: ΔD = D₁ × α × ΔT
Calculation: ΔD = × 10⁻⁵ × 200 ΔD = × 10⁻³ ΔD = 0.0144 cm
Answer: The change in diameter of the hole is 0.0144 cm (or ⁻² cm).
Explanation
This question tests the concept that when a solid sheet with a hole is heated, the hole expands as if it were made of the same material. The material around the hole expands outward, causing the hole to expand proportionally. The linear expansion formula Δl/l = αΔT from the textbook is applied directly to the diameter. The temperature change of 200 K (same magnitude as 200°C) is used with the given coefficient of linear expansion for copper.
Solution Steps
Step 1: Calculate change in temperature: ΔT = 227 - 27 = 200 K
Step 2: Apply linear expansion formula: ΔD = D₁ × α × ΔT
Step 3: Substitute values: ΔD = × 10⁻⁵ × 200
Step 4: Calculate: ΔD = × 10⁻³ = 0.0144 cm