To what temperature must a cylindrical rod of tungsten \(15.025 \mathrm{mm}\) in diameter and a plate of 1025 steel having a circular hole \(15.000 \mathrm{mm}\) in diameter have to be heated for the rod to just fit into the hole? Assume that the initial temperature is \(25^{\circ} \mathrm{C}\).

First, we need to find the coefficients of linear expansion for tungsten and 1025 steel. These values can be found in a reference table or online. The coefficients of linear expansion are: For tungsten: \(\alpha_\text{Tungsten} = 4.5 \times 10^{-6} \frac{1}{\text{°C}}\) For 1025 steel: \(\alpha_\text{Steel} = 10.8 \times 10^{-6} \frac{1}{\text{°C}}\)

The formula for linear expansion is: \(\Delta L = \alpha L_0 \Delta T\) Where \(\Delta L\) is the change in length, \(\alpha\) is the coefficient of linear expansion, \(L_0\) is the initial length (or diameter in this case), and \(\Delta T\) is the change in temperature.

We can now set up equations for the change in diameter of the rod and the hole in the steel plate. For the rod: \(\Delta D_\text{rod} = \alpha_\text{Tungsten} D_0^{rod} \Delta T\) For the hole: \(\Delta D_\text{hole} =\alpha_\text{Steel} D_0^{hole} \Delta T\)

The change in diameter required for the rod to just fit into the hole is: \(\Delta D_\text{rod} - \Delta D_\text{hole} = D_0^{hole} - D_0^{rod}\) Substituting the equations from Step 3, we get: \((\alpha_\text{Tungsten} D_0^{rod} - \alpha_\text{Steel} D_0^{hole}) \Delta T = D_0^{hole} - D_0^{rod}\)

We now solve the equation from Step 4 for \(\Delta T\). Rearrange the equation: \(\Delta T = \frac{D_0^{hole} - D_0^{rod}}{\alpha_\text{Tungsten} D_0^{rod} - \alpha_\text{Steel} D_0^{hole}}\) Plug in the given values: \(\Delta T = \frac{15.000 \text{ mm} - 15.025 \text{ mm}}{(4.5 \times 10^{-6} \frac{1}{\text{°C}})(15.025 \text{ mm}) - (10.8 \times 10^{-6} \frac{1}{\text{°C}})(15.000 \text{ mm})}\) \(\Delta T \approx -557.95^\circ \text{C}\) Since the initial temperature is \(25^\circ \text{C}\), the final temperature would be: \(T_\text{final} = T_0 + \Delta T = 25^\circ \text{C} - 557.95^\circ \text{C} \approx -532.95^\circ \text{C}\) However, finding a negative final temperature does not make physical sense. This indicates an error in our assumption that the rod and hole would expand in a way such that they would fit together. This error arises because the coefficients of expansion are not constant over a large temperature range. Instead, they change with temperature, and the values given are only valid for a small temperature range around \(25^\circ \text{C}\). This means that, with the given values in this exercise, the rod will not fit into the hole simply by increasing the temperature.