question_answer

           One end of a spring of force constant k is fixed to a vertical wall and the other to a block of mass m resting on a smooth horizontal surface. There is another wall at a distance \[{{x}_{0}}\] from the black. The spring is then compressed by \[2{{x}_{0}}\] and released. The time taken to strike the wall is

A)                                                      \[\frac{1}{6}\pi \sqrt{\frac{k}{m}}\]                               

B)            \[\sqrt{\frac{k}{m}}\]

C)            \[\frac{2\pi }{3}\sqrt{\frac{m}{k}}\]                               

D)            \[\frac{\pi }{4}\sqrt{\frac{k}{m}}\]

Correct Answer: C

Solution :

                   The total time from A to C \[{{t}_{Ac}}={{t}_{AB}}+{{t}_{BC}}\]                    \[=(T/4)+{{t}_{BC}}\]                    where T = time period of oscillation of spring mass system                    \[{{t}_{BC}}\] can be obtained from, \[BC=AB\sin (2\pi /T)\,{{t}_{BC}}\]                    Putting \[\frac{BC}{AB}=\frac{1}{2}\] we obtain \[{{t}_{BC}}=\frac{T}{12}\]            Þ \[{{t}_{AC}}=\frac{T}{4}+\frac{T}{12}=\frac{2\pi }{3}\sqrt{\frac{m}{k}}\].