First slide

Thermal expansion

Question

The co-efficient of thermal expansion of a rod is temperature dependent and is given by the formula $α = aT,$ where a is a positive constant and T in $^{0} C .$ If the length of the rod is l at temperature $0^{0} C,$ then the temperature at which the length will be $3 l$ is

Moderate

A

$\sqrt{\frac{\ln 3}{a}}$
B

$\sqrt{\frac{\ln 9}{a}}$
C

$\frac{1}{a}$
D

$\frac{2}{a}$

Solution

dl = l $α$ dT
$\frac{d l}{l} = a T d T \int_{l}^{3 l} \frac{d l}{l} = \int_{0}^{T_{o}} a T . d T {[\ln l]}_{l}^{3 l} = {[\frac{a T^{2}}{2}]}_{0}^{T_{o}} \ln 3 = \frac{a T_{o}^{2}}{2} T_{o} = \sqrt{\frac{\ln 9}{a}}$

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