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

`Y=("Stress")/("strain") = (T//A)/(Delta l//l)`
`T = (Y*Deltal)/(l) A = Y*A alpha Delta T`
In both the rods tension will be same so
`T_(1)=T_(2) ,` Hence, `Y_(1)A_(1)alpha_(1)Delta T = Y_(2)A_(2)alpha_(2)Delta T`
`(A_1)/(A_2) = (Y_(2)alpha_(2))/(Y_(1)alpha_(1))`.