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 $$frequency1$$δkm. The value of δ is $$______$$

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 $$frequency1$$δkm. The value of δ is $$______$$

Infinity Learn · Accepted Answer

Δℓ→x At ℓ:$$Fe=FSPk$$ℓ$$=2kpq$$ℓ$$3Fnet$$ $$=Fsp$$−$$Fe=k($$ℓ$$+x)$$−$$q(2kp)($$ℓ$$+x)3=k(x+$$ℓ$$)$$−$$q(2kp)$$ℓ$$3(1+x/$$ℓ$$)3=$$$$kx$$$$+k$$ℓ−$$q2kp$$ℓ$$31$$−$$3x$$ℓ$$=$$$$kx$$$$+k$$ℓ−$$q2kp$$ℓ$$3+2kpq$$ℓ$$3$$⋅$$3x$$ℓ⇒$$Fnet=$$$$kx$$$$+k$$ℓ$$3x$$ℓ$$=4kxkeq=4k$$           ∴ $$T=2$$$$π$$$$m4k=$$πmk⇒$$f=1$$$$π$$km So, δ$$=$$π$$=3.14$$

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