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 ?

Options

A. `(L_(1))/(L_(2))=(alpha_(2))/(alpha_(1))`
B. `(L_(1))/(L_(2))=sqrt((alpha_(2))/(alpha_(1)))`
C. `(L_(1))/(L_(2))=2(alpha_(2))/(alpha_(1))`
D. `(L_(1))/(L_(2))=2sqrt((alpha_(2))/(alpha_(1)))`

Answer

D

Explanation

Here, h remains constant while the set-up is being heated.
but `L_(2)^(2)=h^(2)+x^(2)`
`2L_(2)DeltaL_(2)=2XDeltaX` for small increment
or `L_(2)DeltaL_(2)=(L_(1))/(2)(DeltaL_(1))/(2)rArr4L_(2)DeltaL_(2)=L_(1)DeltaL_(1)`
`or,4L_(2)^(2)alpha_(2)DeltaT=L_(1)^(2)alpha_(1)DeltaTor,(L_(1))/(L_(2))=2sqrt((alpha_(2))/(alpha_(1)))`