Two rods are joined between fixed supports as shown in the figure. Condition for no change in the length of individual rods with the increase of temperature will be
`"(" alpha_(1), alpha_(2)=` linear expansion coefficient
`A_(1), A_(2)=` Area of rods
`Y_(1), Y_(2)=` Young modulus `")"`

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

One can take an elastic rod as a spring under elastic limits . If the temperature is increased by `Deltatheta` then `L_(1)=L_(1)(l+alpha_(1)Deltatheta),L_(2)(l+alpha_(2)Deltatheta)` where `L_(1)andL_(2)` are Natural. Length of rod at final temperature. By force balance
`K_(1)X_(1)=K_(2)X_(2)or,(Y_(1)A_(1))/(L_(1)).L_(1)alpha_(1)Deltatheta=(Y_(2)A_(2))/(L_(2))alpha_(2)Deltathetaor,Y_(1)A_(1)alpha_(1)=Y_(2)A_(2)alpha_(2),(A_(1))/(A_(2))=(Y_(2)alpha_(2))/(Y_(1)alpha_(1))`