Question

Calculate the compressional force required to prevent the metallic rod of length l cm and cross-sectional area A $c{m}^2$ when heated through $t^{\circ }C$, from expanding along lengthwise. The Young's modulus of elasticity of the metal is E and mean coefficient of linear expansion is $\alpha$ per degree celsius

• A. $EA\alpha t$
• B. $EA\alpha t/(1+\alpha t)$
• C. $EA\alpha t/(1-\alpha t)$
• D. $El\alpha t$

Answer 1

Answer: (B)

$EA\alpha t/(1+\alpha t)$

Thermal expansion in form of change in length (Linear expansion)
$l_2=l_1(1+\alpha t)$

$changeinlength=\left(l\Delta t\right)$

$\Delta t=(t_2-t_1)$

$Youn{g}^{\prime }sModulus=E$

$E=\frac{F}{A}/\frac{l}{L}$

$E=FL/Al$

$L=(l+\Delta l)$

$L=l+\Delta l$

$E=\frac{F}{A}/\frac{l}{L}$

$E=FL/Al$

$E=\frac{F(l+\Delta l)}{A\Delta l}$

$F=EA\Delta l/(l+\Delta l)$

$F=\frac{EA\left(\alpha t\right)}{1+\alpha t}$