Engineering
Physics
SHM of Spring Block System
Electric Dipole
Question

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 p pointing 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 by Δl ≪ l from its equilibrium position and released, it is found to oscillate at frequency 1δkm. The value of δ is ______.

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Solution

l → x

At l : Fe = Fsp

kℓ=2kpq3

Fnet=FspFe=k(+x)q(2kp)(+x)3

=k(x+)q(2kp)3(1+x/)3

=kx+kℓq(2kp3)(13x)

=kx+kℓq(2kp3)+2kpq33x

FN=kx+kℓ(3x)=4kx

k­eq = 4k          T=2πm4k=πmk

f=1πkm

So        δ = π = 3.14