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Thermodynamics

# A composite bar of length $\;(l_{1} + l_{2})\;$ is made from two bars of length $\;l_{1}\;$ and $\;l_{2}\;$ . Their coefficients of linear expansion $\;\alpha_{1}\;$ and $\;\alpha_{2}\;$ respectively . The effective linear coefficient of composite bar

$(a)\;\large\frac{l_{1} \alpha_{1} - l_{2} \alpha_{2}}{l_{1}-l_{2}}\qquad(b)\;\large\frac{l_{1} \alpha_{1} + l_{2} \alpha_{2}}{l_{1}+l_{2}}\qquad(c)\;\alpha_{1}+\alpha_{2}\qquad(d)\;\large\frac{\alpha_{1} + \alpha_{2}}{2}$

Answer : $\;\large\frac{l_{1} \alpha_{1} + l_{2} \alpha_{2}}{l_{1}+l_{2}}$
Explanation :
$\bigtriangleup l_{1} = l_{1} \alpha_{1} \bigtriangleup t$
$\bigtriangleup l_{2} = l_{2} \alpha_{2} \bigtriangleup t$
Total change in length
$\bigtriangleup l_{1} + \bigtriangleup l_{2} = (l_{1} \alpha_{1}+\alpha_{2} l_{2}) \bigtriangleup t$
Therefore , $\;\alpha_{comp}\; = \large\frac{l_{1} \alpha_{1}+l_{2} \alpha_{2}}{l_{1}+l_{2}}$