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A particle, with restoring force proportional to displacement and resistive force proportional to velocity is subjected to a force F sin ω₀t. If the amplitude of the particle is maximum for ω = ω₀ and the energy of the particle is maximum for ω = ω₂, then

1. ω₁ = ω₂ and ω₂ ≠ ω₀
2. ω₁ = ω₀ and ω₂ = ω₀
3. ω₁ ≠ ω₀ and ω₂ = ω₀
4. ω₁ ≠ ω₀ and ω₂ ≠ ω₀

Correct answer: ω₁ = ω₀ and ω₂ = ω₀

Solution

The amplitude of a driven damped harmonic oscillator is maximum at the driving frequency ω₀, which matches the natural frequency of the system. The energy of the particle is also maximum at this resonance frequency, ω₀. Hence, ω₁ = ω₀ and ω₂ = ω₀.

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