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 δ1mk. The value of δ is ____.
Answer: 3.14
Solution
At equilibrium: k ℓ = 2kpq/ℓ³. When displaced by x: F_net = k(ℓ+x) - 2kpq/(ℓ+x)³ = kx + kℓ - (2kpq/ℓ³)(1 - 3x/ℓ) = kx + kℓ - kℓ + 3kℓ(x/ℓ) = 4kx. Thus k_eq = 4k, giving frequency f = (1/2π)√(4k/m) = (1/π)√(k/m). Comparing with (1/δ)√(k/m): δ = π ≈ 3.14.
