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

`T_f=399.446\ ^(\circ)C`

Step-by-step 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* 10^(-6)\ ^(\circ)C^(-1)`

• coefficient of linear expansion of plate,
  `\alpha_p=17* 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* 10.034 +\pi* 10.034 * (3.5* 10^(-6))* (T_f-28)=\pi* 9.984 +\pi* 9.984 * (17* 10^(-6))* (T_f-28)`

`T_f=399.446\ ^(\circ)C`