Question

A metal rod of length ‘$L$’ and cross-sectional area ‘$A$’ is heated through $‘T’ \, {^{\circ}C}$. What is the force required to prevent the expansion of the rod lengthwise ?
[$Y =$ Young's modulus of the material of rod , $\alpha$ = coefficient of linear expansion]

• A. $\frac{Y A \alpha T}{(1 - \alpha T)}$
• B. $\frac{Y A \alpha T}{(1 + \alpha T)}$
• C. $\frac{(1 - \alpha T)}{Y A \alpha T}$
• D. $\frac{(1 + \alpha T)}{Y A \alpha T}$

Answer 1

Answer: (B)

change in length $\delta L = L \alpha T$
final length $L _{ f }= L +\delta L$
now in order to to constrain the expansion,
there should be force that should compress the rod by $\delta L$
strain $=\frac{\delta L }{ L _{ f }}$
$Y =\frac{ F / A }{\text { strain }}$
$F =\frac{ YA A T }{1+\alpha T }$