question_answer

A small sphere carrying a charge q is having in between two parallel plates by a string of length L. Time period of pendulum is \[{{T}_{0}}\]. When parallel plates are charged, the time period changes to T, the ratio of \[\operatorname{T} / {{T}_{0}}\] is equal to

A)  \[{{\left( \frac{g+\frac{qE}{m}}{g} \right)}^{1/2}}\]                 

B)  \[{{\left( \frac{g}{g+\frac{qE}{m}} \right)}^{3/2}}\]

C)  \[{{\left( \frac{g}{g+\frac{qE}{m}} \right)}^{1/2}}\]                 

D)  None of these

Correct Answer: D

Solution :

Time period of simple pendulum when plates are not changed. \[{{T}_{0}}=2\,\pi \sqrt{\frac{L}{g}}\]where, \[\operatorname{L} = length of string and\]\[\operatorname{g} =acceleration due to gravity\]   When plates are charged electric field E will appear. \[\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{F}_{E}}= qE = mg\] where q is charge on sphere. \[\therefore \] Resultant acceleration is \[g'=g+\frac{qE}{m}\] \[\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,T=2\pi \frac{g}{g+\frac{qE}{m}}\]