Question

To what temperature (in °C) must a cylindrical rod of one metal 10.034 mm in diameter and a plate of second metal having a circular hole 9.984 mm in diameter have to be heated for the rod to just fit into the hole? Assume that the initial temperature is 28°C and that the linear expansion coefficient values for metals one and two are 3.5 x 10-6 (°C)-1 and 17 x 10-6 (°C)-1, respectively.

Answer 1

Answer:

$T_f=399.446\ ^{\circ}C$

Explanation:

Given:

• diameter of a rod, $d_r=10.034\ mm$

• diameter of hole on the plate, $d_h=9.984\ mm$

• initial temperature, $T_i=28^{\circ}C$

• coefficient of linear expansion of rod, $\alpha_r=3.5\times 10^{-6}\ ^{\circ}C^{-1}$

• coefficient of linear expansion of plate,  $\alpha_p=17\times 10^{-6}\ ^{\circ}C^{-1}$

Now using the equation of linear expansion:

$\pi.d_r+\pi.d_r.\alpha_r.(T_f-T_i)=\pi.d_h+\pi.d_h.\alpha_p.(T_f-T_i)$

$\pi\times 10.034 +\pi\times 10.034 \times (3.5\times 10^{-6})\times (T_f-28)=\pi\times 9.984 +\pi\times 9.984 \times (17\times 10^{-6})\times (T_f-28)$

$T_f=399.446\ ^{\circ}C$