A glass window is to be installed in an aluminium frame. On the working day, the temperature is 40^ C and the glass window measures exactly 20 cm 30 cm . Determine the required size of the aluminium frame so that no stress is exerted on the glass in winter, even if the temperature falls to 0^ C . The coefficients of linear expansion for glass and aluminium are 9.0 10⁻⁶ ^ C⁻¹ and 24 10⁻⁶ ^ C⁻¹ , respectively.

Step-by-step solution

Let the dimensions of the glass at 40^ C be L_ g1 = 20 cm and L_ g2 = 30 cm . Let the dimensions of the aluminium frame at 40^ C be L_ a1 and L_ a2 . When the temperature falls to 0^ C , the drop in temperature is T = 40^ C . To ensure no stress is exerted on the glass, the dimensions of the aluminium frame at 0^ C must be equal to the dimensions of the glass at 0^ C . L_ a1 (1 - _a T) = L_ g1 (1 - _g T) L_ a1 = L_ g1 1 - _g T 1 - _a T L_ g1 [1 + ( _a - _g) T] Substituting the given values: _a - _g = (24 - 9) 10⁻⁶