A pendulume clock loses $12\;s$ a day if the temperature is $40^oC$ and gains $4\;s$ a day if the temperature is $20^oC$. The temperature at which the clock will show correct time, and the coeffecient of linear expansion $(\alpha)$ of the metal of the pendulum shaft are respectively
JEE MAIN 2016, Diffcult
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$\mathrm{T}=2 \pi \sqrt{\frac{\ell}{\mathrm{g}}}$
$\frac{\Delta \mathrm{T}}{\mathrm{T}}=\frac{1}{2} \frac{\Delta \ell}{\ell}$
When clock gain $12\, sec$
$\frac{12}{\mathrm{T}}=\frac{1}{2} \alpha(40-\theta)$ $...(1)$
When clock lose $4\, sec.$
$\frac{4}{\mathrm{T}}=\frac{1}{2} \alpha(\theta-20)$ $...(2)$
From equation $( 1)\&(2)$
$3=\frac{40-\theta}{\theta-20}$
$3 \theta-60=40-\theta$
$4 \theta=100$
$\theta=25^{\circ} \mathrm{C}$
from equation $( 1 )$
$\frac{12}{\mathrm{T}}=\frac{1}{2} \alpha(40-25)$
$\frac{12}{24 \times 3600}=\frac{1}{2} \alpha \times 15$
$\alpha=\frac{24}{24 \times 3600 \times 15}$
$\alpha=1.85 \times 10^{-15} /^{\circ} \mathrm{C}$
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