The focal length of a spherical mirror made of steel is \(150 \mathrm{~cm}\). If the temperature of the…
The focal length of a spherical mirror made of steel is \(150 \mathrm{~cm}\). If the temperature of the mirror increases by \(200 \mathrm{~K}\), its focal length become (coefficient of linear expansion of steel \(\alpha=12 \times 10^{-60} \mathrm{C}^{-1}\).)
\(186.3 \mathrm{~cm}\)
\(153.6 \mathrm{~cm}\)
\(150.036 \mathrm{~cm}\)
\(150.36 \mathrm{~cm}\)
Solution
Given,
focal length of spherical mirror, \(f=150 \mathrm{~cm}\) coefficient of linear expansion of steel,
\(\alpha=12 \times 10^{-6 \circ} \mathrm{C}^{-1}\)
As we know that, coefficient of linear expansion, \(\alpha=\frac{\Delta R}{R \times \Delta T}\)
\(\begin{aligned} \frac{\Delta R}{R} & =\alpha \Delta T \\ \frac{\Delta f}{f} & =\alpha \Delta T \quad\left(\because f=\frac{R}{2}\right) \\ \Delta f & =f \alpha \Delta T \\ f^{\prime}-f & =f \alpha \Delta T\end{aligned}\)
where, \(f=\) initial focal length of the mirror and \(f^{\prime}=\) final focal length of mirror
\(f^{\prime}=f+f \alpha \Delta T\)
\(f^{\prime}=f(1+\alpha \Delta T)\)
\(f^{\prime}=150\left(1+12 \times 10^{-6} \times 200\right) \quad(\because \Delta T=200 \mathrm{~K})\)
\(f^{\prime}=150.36 \mathrm{~cm}\)