Two rods of different materials having coefficients of linear expansion α1

Question

Two rods of different materials having coefficients of linear expansion

α_{1}

and

α_{2}

and Young’s modulus

Y_{1}

and

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 rods. If

α_{1}

α_{2}

= 2:3, the thermal stress developed in the two rods are equal provided

Y_{1} : Y_{2}

equal to

Options:

A . 2 : 3

B . 4 : 9

C . 1 : 2

D . 3 : 2

Answer: Option D
:
D
From the formula for thermal stress

(F / A)_{1} = α_{1} y_{1} △ T

(F / A)_{2} = α_{2} y_{2} △ T

Since,temperature

△ T

is same

\frac{(\frac{F}{A})_{1}}{(\frac{F}{A})_{2}} = \frac{α_{1} y_{1}}{α_{2} y_{2}}

For thermal stress to be equal

α_{1} y_{1} = α_{2} y_{2}

o r \frac{y_{1}}{y_{2}} = \frac{α_{2}}{α_{1}} = \frac{3}{2}

since

\frac{α_{1}}{α_{2}} = 2 : 3)

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