First slide

Thermal expansion

Question

Two wires A and B of the same cross sectional areas, Young's moduli $Y_{1} and Y_{2}$ and coefficients of linear expansion $α_{1} and α_{2}$ respectively are joined together and fixed between rigid supports at either ends. The tension in the compound wire when wire A is heated and wire B is cooled at different temperature is same when wire A alone is cooled at the same temperature as wire B earlier. Find correct option.

Difficult

A

$\frac{α_{1}}{α_{2}} > \frac{Y_{2}}{2 Y_{1}}$
B

$\frac{α_{1}}{α_{2}} < \frac{Y_{2}}{2 Y_{1}}$
C

$\frac{α_{1}}{α_{2}} > \frac{2 Y_{2}}{Y_{1}}$
D

$\frac{α_{1}}{α_{2}} > \frac{Y_{2}}{Y_{1}}$

Solution

Tension in any wire, F = YA $α$ t
Given $Y_{2} A_{2} α_{2} t_{2} - Y_{1} A_{1} α_{1} t_{1} = Y_{1} A_{1} α_{1} t_{2}$
$Y_{2} A_{2} α_{2} t_{2} = Y_{1} A_{1} α_{1} (t_{1} + t_{2})$
   $\frac{(t_{1} + t_{2})}{t_{2}} = \frac{Y_{2} A_{2} α_{2}}{Y_{1} A_{1} α_{1}}$
or    $\frac{t_{1} - t_{2}}{t_{2}} = \frac{Y_{2} A_{2} α_{2} - 2 Y_{1} A_{1} α_{1}}{Y_{1} A_{1} α_{1}}$
   $\frac{∆ t}{t_{2}} = \frac{Y_{2} α_{2} - 2 Y_{1} α_{1}}{Y_{1} α_{1}} (since A_{1} = A_{2})$
For $∆ t > 0$
$Y_{2} α_{2} > 2 Y_{1} α_{1}$
or $\frac{Y_{2}}{2 Y_{1}} > \frac{α_{1}}{α_{2}}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App