Show that the coefficient of superficial expansion of a rectangular sheet of the solid is twice its coefficient of linear expansion.

Explanation

Let `a_(2),b_(1)` be the length and breadth, respectively, of the rectangular sheet at the initial temeprature `t_(1)`.
Then, `S_(1)=a_(1)b_(1)=` initial area
Now, the temperature is raised to `t_(2), " where " t_(2)-t_(1)=t`.
The new values of the length and breadth are, respectively, `a_(2) " and " b_(2), " area " S_(2)=a_(2)b_(2)`.
If `alpha` be the coefficient of linear expansion, then
`a_(2)=a_(1)(1+alphat) " and " b_(2)=b_(1)(1+alphat)`
`therefore S_(2)=a_(2)b_(2)=a_(1)b_(1)(1+alphat)^(2)=S_(1)(1+2alphat)`
`" "` [ neglecting `alpha^(2)t^(2)` as `alpha "<<" 1`]
On the other hand, if `beta` be the coefficient of superficial expansion, then
`S_(2)=S_(1)(1+betat)`
On comparison, we have `beta=2alpha`.