question_answer 36)

                  When a mass m is connected individually to two springs \[{{S}_{1}}\] and \[{{S}_{2}}\], the oscillation frequencies are \[{{v}_{1}}\] and \[{{v}_{2}}\]. If the same mass is displacement 0 1 2 3 4 5 6 7 time (s) attached to the two springs as shown in Fig., the oscillation frequency would be (a) \[{{v}_{1}}+{{v}_{2}}\]              (b) \[\sqrt{v_{1}^{2}+v_{2}^{2}}\] (c) \[{{\left( \frac{1}{{{v}_{1}}}+\,\frac{1}{{{v}_{2}}} \right)}^{-1}}\]                    (d) \[\sqrt{v_{1}^{2}-v_{2}^{2}}\]

Answer:

                  (b) \[{{v}_{1}}\,=\frac{1}{2\pi }\,\sqrt{\frac{{{s}_{1}}}{m}}\] and \[{{v}_{2}}\,=\,\frac{1}{2\pi }\,\,\sqrt{\frac{{{s}_{2}}}{m}}\]                 When connected as shown,  \[s={{s}_{1}}+{{s}_{2}}\]                 \[\therefore \] \[v=\,\frac{1}{2\pi }\,\,\sqrt{\frac{{{s}_{1}}+\,{{s}_{2}}}{m}}\]                 \[=\,\frac{1}{2\pi }\,\sqrt{\frac{{{s}_{1}}}{m}+\,\frac{{{s}_{2}}}{m}}\,=\frac{1}{2\pi }\,\sqrt{4{{\pi }^{2}}v_{1}^{2}\,+4{{\pi }^{2}}v_{2}^{2}}\]                 \[=\sqrt{v_{1}^{2}+\,v_{2}^{2}}\]