A smooth fixed rod is inclined at angle θ to the horizontal. A fixed charge +Q is at its lower end. A bead of mass m and charge q can slide freely on the rod. Its equilibrium distance from Q is x₀. The angular frequency of small oscillations about equilibrium is:
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Question

A smooth fixed rod is inclined at angle θ to the horizontal. A fixed charge +Q is at its lower end. A bead of mass m and charge q can slide freely on the rod. Its equilibrium distance from Q is x₀. The angular frequency of small oscillations about equilibrium is:

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G1 Physics · Accepted Answer

Along the rod, equilibrium requires kQq/x₀² = mg sinθ. If the bead is displaced by a small distance ξ, kQq/(x₀+ξ)² ≈ kQq/x₀² - 2kQq ξ/x₀³. The constant part cancels gravity, leaving mξ¨ = -(2kQq/x₀³)ξ. Hence, ω² = 2kQq/(mx₀³). Using the equilibrium condition, ω² = 2g sinθ/x₀, so ω = √[2g sinθ/x₀].