Question

A brass lid screws tightly onto a glass jar at 20 degrees C. To help open the jar, it can be placed into a bath of hot water. After this treatment, the temperature of the lid and the jar are both 60 degrees C. The inside diameter of the lid is 8.0 cm at 20 degrees C. Find the size of the gap (difference in radius) that develops by this procedure.

Answer 1

Answer:

0.0016 cm

Explanation:

$\alpha_b$ = Thermal coefficient of expansion of brass = $19\times 10^{-6}\ /^{\circ}C$

$\alpha_g$ = Thermal coefficient of expansion of glass = $9\times 10^{-6}\ /^{\circ}C$

$\Delta T$ = Change in temperature = $(60-20)^{\circ}C$

$R_0$ = Initial radius = 4 cm

Change in radius of material is given by

$R=R_0(1+\alpha\Delta T)$

Difference in radii of the lid and jar

$\Delta R=R_b-R_g\\\Rightarrow \Delta R=R_0(1+\alpha_b\Delta T)-R_0(1+\alpha_g\Delta T)\\\Rightarrow \Delta R=R_0(\alpha_b-\alpha_g)\Delta T\\\Rightarrow \Delta R=4\times (19\times 10^{-6}-9\times 10^{-6})\times (60-20)\\\Rightarrow \Delta R=0.0016\ cm$

The size of the gap is 0.0016 cm or 0.000016 m