Question

A glass window is to be fit in an aluminium frame. The temperature on the working day is 40°C and the glass window measures exactly 20 cm × 30 cm. What should be the size of the aluminium frame so that there is no stress on the glass in winter even if the temperature drops to 0°C? Coefficients of linear  expansion for glass  and aluminium are 9.0 × 10^–6 °C^–1 and 24 ×100^–6°C^–1 , respectively.

Answer 1

Given:
At 40^oC, the length and breadth of the glass window are 20 cm and 30 cm, respectively.
Coefficient of linear expansion of glass, 

\[\alpha_g\] =  9.0 × 10^–6 °C^–1

Coefficient of linear expansion for aluminium,

\[\alpha_{Al}\] = 24 ×100^–6 °C^–1

The final length of aluminium should be equal to the final length of glass so that there is no stress on the glass in winter, even if the temperature drops to 0 °C.
​Change in temperature,

\[\Delta\theta\]  = 40 °C

Let the initial length of aluminium be l.

\[l\left( 1 - \alpha_{Al} ∆ \theta \right) = 20\left( 1 - \alpha_g ∆ \theta \right)\]

\[ \Rightarrow l\left( 1 - 24 \times {10}^{- 6} \times 40 \right) = 20\left( 1 - 9 \times {10}^{- 6} \times 40 \right)\]

\[ \Rightarrow l\left( 1 - 0 . 00096 \right) = 20\left( 1 - 0 . 00036 \right)\]

\[ \Rightarrow l = \frac{20 \times 0 . 99964}{1 - 0 . 00096}\]

\[ = \frac{20 \times 0 . 99964}{0 . 99904}\]

\[ \Rightarrow l = 20 . 012 cm\]

Let the initial breadth of aluminium be b.

\[b\left( 1 - \alpha_{Al} ∆ \theta \right) = 30\left( 1 - \alpha_g ∆ \theta \right)\]

\[ \Rightarrow b = \frac{30 \times \left( 1 - 9 \times {10}^{- 6} \times 40 \right)}{\left( 1 - 24 \times {10}^{- 6} \times 40 \right)}\]

\[ = \frac{30 \times 0 . 99964}{0 . 99904}\]

\[ \Rightarrow b = 30 . 018 cm\]

Therefore, the size of the aluminium frame should be 20.012 cm × 30.018 cm.