Question
Calculate focal length of a spherical mirror from the following observations: object distance $\text{u} = (50.1 \pm 0.5)\text{cm}$ and image distance $\text{v} = (20.1 \pm0.2)\text{cm} .$
✓
Answer
As $\frac{1}{\text{f}}=\frac{1}{\text{u}}+\frac{1}{\text{v}}=\frac{\text{v}+\text{u}}{\text{uv}}$ $\therefore\text{f}=\frac{\text{uv}}{\text{u}+\text{v}}=\frac{(50.1)(20.1)}{(50.1+20.1)}=14.3\text{cm}$ Also, $\frac{\Delta\text{f}}{\text{f}}=\pm\Big[\frac{\Delta\text{u}}{\text{u}}+\frac{\Delta\text{v}}{\text{v}}+\frac{\Delta\text{u}+\Delta\text{v}}{\text{u}+\text{v}}\Big]$ $=\pm\Big[\frac{0.5}{50.1}+\frac{0.2}{20.1}+\frac{0.5+0.2}{50.1+20.1}\Big]$ $\frac{\Delta\text{f}}{\text{f}}=\pm\Big[\frac{1}{100.2}+\frac{1}{100.5}+\frac{0.7}{70.2}\Big]$ $=\pm[0.0098+0.00995]+0.00997]$ $\Delta\text{f}=0.02990\times\text{f}=0.0299\times14.3$ $=0.428\text{cm}=0.4\text{cm}$ $\therefore\text{f}=(14.3\pm0.4)\text{cm}$
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