question_answer

Calculate focal length of a spherical mirror from the following observations : object distance \[\text{u}=\left( \text{5}0.\text{1}\pm 0.\text{5} \right)\text{cm}\] and image distance, \[\text{v}=\left( \text{2}0.\text{1}\pm 0.\text{2} \right)\text{cm}\].

Answer:

                                                                                                                  As      \[\frac{1}{f}=\frac{1}{u}=\frac{1}{v}=\frac{v+u}{uv}\]                                         \[f=\frac{uv}{u+v}=\frac{(50.1)(20.1)}{(50.1+20.1)}=\text{ 14}.\text{3 cm}\] . From                                                                    \[\frac{1}{f}=\frac{1}{u}+\frac{1}{v}\]                                           \[\frac{-\Delta f}{{{f}^{2}}}=\frac{-\Delta u}{u}-\frac{\Delta v}{{{v}^{2}}}\] \[\Delta \text{f }=\Delta \text{u}{{\left( \frac{f}{u} \right)}^{2}}+\Delta \upsilon {{\left( \frac{f}{\upsilon } \right)}^{2}}=0.5{{\left( \frac{14.3}{20.1} \right)}^{2}}=0.04+0.10=0.14cm\]                                                \[\text{f }=\left( \text{14}.\text{3 }\pm 0.\text{1} \right)\text{ cm}\] .