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 attached at its other end . The entire system is kept on a smooth horizontal surface . When a point dipole p pointing towards the charge q is fixed at the origin , the spring gets stretched to a length l and attains a new equilibrium positition (see figure below) . if the point mass is now displaced slightly by `Deltal ltltl` from its equilibrium position and released it is found to oscillate at frequency `(1)/(delta)sqrt((k)/(m))`. The value of `delta` is ________.

Answer

`03.14`

Explanation

Original frequency `f=(2)/(lamda)sqrt((k)/(m))`
if dipole appears
At equilibrium
`(2kp)/((l+x_(0))^(3)).qx_(0)` . . . . .(i)

Now disturbed
`vec(x)(2kp)/((l+x_(0)+x)^(3)).q-k(x_(0)+x)=m`
`ma=(2kpq)/((l+x_(0))^(3))[1+(x)/(l+x_(0))]^(-3)-k(x_(0)+x)`
`=(2kpq)/((l+x_(0))^(3))[1-(3x)/(l+x_(0))kx_(0)-kx=-kx[(3x_(0))/(l+x_(0))+1]`
As l is negative
`ma=-kx.4," "a=(4k)/(m)`
New frequency :
`f.=(1)/(2pi)sqrt((4k)/(m))=2f" "=1/pi sqrt((k)/(m))`
`lamda=pi=3.14`.