Two rods of different materials having coefficients of linear expansion ${\alpha }_1, {\alpha }_2$ and Young’s moduli $Y_1, Y_2$ respectively are fixed between two rigid massive walls. The rods are heated such that they undergo the same increase in temperature. There is no bending of the rods. If ${\alpha }_1 : {\alpha }_2 = 2 : 3$ the thermal stresses developed in the two rods are equal, provided $Y_1 : Y_2$ is equal to :

A wooden wheel of radius $R$ is made of two semi-circular parts (see figure). The two parts are held together by a ring made of a metal strip of cross-sectional area $A$ and length $L$. $L$ is slightly less than $2nR$. To fit the ring on the wheel, it is heated so that its temperature rises by $∆T$ and it just steps over the wheel. As it cools down to the surrounding temperature, it presses the semi-circular parts together. If the coefficient of linear expansion of the metal is a and it's Young’s modulus is $Y$, the force that one part of the wheel applies on the other part is :