Question

The coefficient of linear expansion for some common materials are listed in the table to the right. You would like to design an emergency overheating alarm using one of these materials. In your design, at room temperature (23.0°C), a 7.01 cm long cylinder of the material will sit with one end against a wall and the other just barely touching a button. Once the heat causes the cylinder to lengthen by 0.0268 cm, the button will be activated, turning on the alarm.
A) At what temperature will the alarm be triggered if the cylinder is made out of polycarbonate? _____ degrees C

B) At what temperature will the alarm be triggered if the cylinder is made out of cast iron? ______ degrees

Answer 1

Answer:

A)    $T_{f}$ = 77.6  ⁰C ,  B) $T_{f}$ = 341.6 C

Explanation:

The thermal expansion is given by

           ΔL = α L₀ ΔT

The thermal expansion coefficient for polycarbonate varies between 65 and 70 10⁻⁶ ⁰C⁻¹ and for cast iron 12 10⁻⁶ C⁻¹

Let's look for the final temperature that says same shift change

          ΔT = ΔL / α L₀

          $T_{f}$ - T₀ = ΔL /α L₀

          $T_{f}$= T₀ + ΔL / α L₀

Let's calculate for each material

A) Polycarbonate use as thermal expansion coefficient 70 10⁻⁶ C⁻¹

         $T_{f}$ = 23.0 + 0.0268 / (70 10⁻⁶ 7.01)

         $T_{f}$ = 23.0 + 54.6

        $T_{f}$ = 77.6  ⁰C

B) cast iron with thermal expansion coefficient 12 10⁻⁶ C⁻¹

      $T_{f}$ = 23.0 + 0.0268 / (12 10⁻⁶ 7.01)

      $T_{f}$ = 23.0 + 3.1859 10²

      $T_{f}$ = 341.6 C