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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 dipoleppointing 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 frequency1δkm. The value of δ is


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Solution

Let us extend the spring x distance from its mean point .
Δlx
At l : Fe =Fsp

kl = 2kpql3

The net force at displaced position :

Fnet=FspFe=k(l+x)q(2kp)(l+x)3
Fnet=k(x+l)q(2kp)l3(1+xl)3

Fnet=kx + klq(2kpl3)(13xl)

Fnet=kx+klq(2kpl3)+2kpql3.3xl
Fnet=kx+kl3xl=4kx
keq=4k
T=2πm4k=πmk
f=1πkm
So,δ=π=3.14

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