Two elastic rods are joined between fixed supports as shown in figure. Condition for no change in the lengths of individual rods with the increase of temperature `(alpha_(1),alpha_(2)`linear expansion coefficient, `A_(1),A_(2)` =area of rods, `Y_(1),Y_(2)` =Young’s modulus) is:

Options

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

Answer

D

Explanation

`Y=("Stress")/("Strain")=(T//A)/(Deltal//l)`
`T=(Y.Deltal)/l A=Y.A alpha DeltaT`
In both the rods tension will be same so
`T_(1)=T_(2)`
Hence `Y_(1)A_(1)alpha_(1)DeltaT=Y_(2)A_(2)alpha_(2)DeltaT`
`(A_(1))/(A_(2))=(Y_(2)alpha_(2))/(Y_(1)alpha_(1))`