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 = 1.70 × \(10^{-5} K^{-1}.\)

Question

The apparent coefficient of expansion of a liquid, when heated in a copper vessel, is \( C \) and when heated in a silver vessel is \( S \). If \( A \) is the linear coefficient of expansion of copper, then the linear coefficient of expansion of silver is:

Answer 1

Given

Initial diameter of hole: d_0 = 4.24
Initial temperature: T_1 = 27.0
Final temperature: T_2 = 227
Coefficient of linear expansion of copper: \( \alpha = 1.70 \times 10^{-5}\,\text{K}^{-1} \)

Concept

A hole in a metal sheet expands as if it were made of the same material as the sheet.
So the diameter of the hole increases with the same linear expansion formula as a rod of copper.

1. Temperature change

\( \Delta T = T_2 - T_1 = 227 - 27 = 200\,\text{K} \)

2. Change in diameter

Linear expansion formula for length (or diameter):

\( \Delta d = \alpha \, d_0 \, \Delta T \)

Substitute values:

\( \Delta d = 1.70 \times 10^{-5} \times 4.24 \times 200 \,\text{cm} \) \( = 1.70 \times 10^{-5} \times 848 \,\text{cm} \) \( = 1.4416 \times 10^{-2} \,\text{cm} \approx 1.44 \times 10^{-2} \,\text{cm} \)

Change in diameter of the hole: \( \Delta d \approx 1.44 \times 10^{-2} \,\text{cm} = 0.0144 \,\text{cm} \) (increase).

3. Final diameter (optional)

\( d_{\text{final}} = d_0 + \Delta d \approx 4.24 + 0.0144 = 4.2544 \,\text{cm} \)