Question

The coefficient of thermal expansion of a rod is temperature-dependent and is given by the formula $\alpha=a T$, where $a$ is $3.465 \times 10^{-5} \quad{ }^{\circ} C ^{-2}$ and $T$ in ${ }^{\circ} C$. If the length of the rod is $l$ at temperature $0^{\circ} C$, then the temperature at which the length will be $2 l$ is [Given, $\ln (2)=0.693]$

• A. $12{ }^{\circ} C$
• B. $20{ }^{\circ} C$
• C. $200{ }^{\circ} C$
• D. $100{ }^{\circ} C$

Answer 1

Answer: (C)

As; $dl=\alpha ldT$
$\therefore \int\limits _{l}^{2 l}\frac{d l}{l}=a \, \int\limits _{0}^{T}TdT$
$ln2=a\frac{T^{2}}{2}$
$\therefore T =\left[\frac{\ln 4}{ a }\right]^{\frac{1}{2}}=\left[\frac{2 \times 0.693}{3.465 \times 10^{-5}}\right]^{\frac{1}{2}}=\sqrt{\frac{2}{5} \times 10^{5}}=200{ }^{\circ} C$