A glass window is to be fit in an aluminum frame. The temperature on the working day is `40^@C` and the glass window measures exactly ` 20 cm xx 30 cm.` What should be the size of the aluminum 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 aluminum are `9.0 xx 10^(-6)C^(-1)` and ` 24xx10^(-6)C^(-1)` respectively.

Explanation

The final length of Aluminium should be
equal to the final length of Glass
Let the initial length of Aluminium `= l`
`l(1-alpha_Al Delta theta)= 20(1-alpha_g Delta theta)`
`rArr (l(1-24 xx 10^-6 xx 40))`
`=20(1-9xx10^-6 xx 40)`
`rArr (1-0.00096)=20(1-0.0036)`
`rArr l=(20 xx 0.99964)/(1-0.00096)`
`=20 xx 0.99964/0.99904 =20.012 cm`
Let initial breadth of Aluminium `= b
`b(1-alpha_Al Delta theta) = 30(1-alpha_g Delta theta)`
`rArr b = 30 xx (1-9 xx 10^-6 xx 40)/(1-24 xx 10^6 xx 40) `
`=30 xx 0.99964/0.99904`
`=30.018 cm`