Two rods of different materials are of the same length and cross-sectional area. They are joined lengthwise and rigidly fixed between two walls. The materials of the two rods have different thermal and mechanical properties. What should be the relation between the Young's moduli and the coefficients of linear expansion of the two materials so that the position of the junction point of the rods does not alter even when heated?

Explanation

The rods are rigidly fixed. If the thermal stresses developed remains the same for all temperatures, the position of the junction point will not change.
Let the Young.s moduli and coefficients of linear expansion for the two materials be `Y_(1), alpha_(1) " and " Y_(2), alpha_(2)` respectively and the temperature be increased by t.
Hence, the thermal stress in the 1st rod `=Y_(1)alpha_(1)t` and that in the 2nd rod `=Y_(2)alpha_(2)t.`
For no movement of the junction point,
`" "Y_(1)alpha_(1)t=Y_(2)alpha_(2)t`
or, `" "Y_(1)/Y_(2)=alpha_(2)/alpha_(1)` is the required relation.