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A metallic rod $1\,cm$ long with a square cross-section is heated through $1^o C$. If Young’s modulus of elasticity of the metal is $E$ and the mean coefficient of linear expansion is $\alpha$ per degree Celsius, then the compressional force required to prevent the rod from expanding along its length is :(Neglect the change of cross-sectional area)
$EA\alpha t$
$EA\alpha t/(1 + \alpha t)$
$EA\alpha t/(1\alpha t)$
$E/\alpha t$
Solution
Thermal expansion in form of change in length (Linear expansion)
$l_{2}=l_{1}(1+\alpha t)$
change in length $=(l \Delta t)$
$\Delta t=\left(t_{2}-t_{1}\right)$
Young's Modulus $=E$
$E=\frac{F}{A} / \frac{l}{L}$
$E=F L / A l$
$L=(l+\Delta l)$
$L=l+\Delta l$
$E=\frac{F}{A} / \frac{l}{L}$
$E=F L / A l$
$E=\frac{F(l+\Delta l)}{A \Delta l}$
$F=E A \Delta l /(l+\Delta l)$
$F=\frac{E A(\alpha t)}{1+\alpha t}$
