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

\(\frac{\Delta \mathrm{T}}{\mathrm{T}}=\frac{1}{2} \frac{\Delta \ell}{\ell}\)

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 \(( 1 )\)

\(\frac{12}{\mathrm{T}}=\frac{1}{2} \alpha(40-25)\)

\(\frac{12}{24 \times 3600}=\frac{1}{2} \alpha \times 15\)

\(\alpha=\frac{24}{24 \times 3600 \times 15}\)

\(\alpha=1.85 \times 10^{-15} /^{\circ} \mathrm{C}\)