Two rods each of length `L_(2)` and coefficient of linear expansion `alpha_(2)` each are connected freely to a third rod of length `L_(1)` and coefficient of expansion `alpha_(1)` to form an isoscles triangle. The arrangement is supported on a knife-edge at the midpoint of `L_(1)` which is horizontal. what relation must exist between `L_(1)` and `L_(2)` so that the apex of the isoscles triangle is to remain at a constant height from the knife edge as the temperature changes ?

Answer

`4L_(2)^(2)alpha_(2) = L_(1)^(2)alpha_(1)`

Explanation

`h^(2) = L_(2)^(2) - (l_(1)^(2))/(4)` = constant
`implies 2L_(2) DeltaL_(2) - (2L_(1)DeltaL_(1))/(4) = 0 `
`implies 4L_(2) (L_(2)alpha_(2)DeltaT) = L_(1)(L_(1)alpha_(1)DeltaT)`
`implies 4L_(2)^(2)alpha_(2) = L_(1)^(2)alpha_(1)`