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Heat and Temperature — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsHeat and Temperature1 Mark
Question
A steel wire of cross-sectional area $0.5 \mathrm{~mm}^2$ is held between two fixed supports. If the wire is just taut at $20^{\circ} \mathrm{C}$, determine the tension when the temperature falls to $0^{\circ} \mathrm{C}$. Coefficient of linear expansion of steel is $1.2 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}$ and its Young's modulus is $2.0 \times 10^{11} \mathrm{~N} \mathrm{~m}^{-2}$.
✓

Answer


$\text{A}=0.5\text{mm}^2=0.5\times10^{-6}\text{m}^2$
$\text{T}_1=20^\circ\text{C},$
$\text{T}_2=0^\circ\text{C}$
$\alpha_\text{s}=1.2\times10^{-5}\ /^\circ\text{C},$
$\text{Y}=2\times2\times10^{11}\text{N/m}^2$
Decrease in length due to compression $=\text{L}\alpha\Delta\theta\ ...(1)$
$\text{Y}=\frac{\text{stress}}{\text{strain}}=\frac{\text{F}}{\text{A}}\times\frac{\text{L}}{\Delta\text{L}}$
$\Rightarrow\Delta\text{L}=\frac{\text{FL}}{\text{AY}}\ ...(2)$
Tension is developed due to (1) & (2)
Equating them,
$\text{L}\alpha\Delta\theta=\frac{\text{FL}}{\text{AY}}$
$\Rightarrow\text{F}=\alpha\Delta\theta\text{AY}$
$=1.2\times10^{-5}\times(20-0)\times0.5\times10^{-5}2\times10^{11}=24\text{N}$

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