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# One end of a spring of negligible unstretched length and spring constant k is fixed at the origin (0,0). A point particle of mass m carrying a positive charge q is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole→ppointing towards the charge q is fixed at the origin, the spring gets stretched to a length l and attains a new equilibrium position (see figure below). If the point mass is now displaced slightly byΔl<<l from its equilibrium position and released, it is found to oscillate at frequency1δ√km. The value of δ is

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Solution

## Let us extend the spring x distance from its mean point . Δl→x At l : Fe =Fsp kl = 2kpql3 The net force at displaced position : Fnet=Fsp−Fe=k(l+x)−q(2kp)(l+x)3 Fnet=k(x+l) – q(2kp)l3(1+xl)3 Fnet=kx + kl−q(2kpl3)(1−3xl) Fnet=kx+kl−q(2kpl3)+2kpql3.3xl Fnet=kx+kl3xl=4kx keq=4k T=2π√m4k=π√mk f=1π√km So,δ=π=3.14

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