#SHM

Forced Oscillations and Resonance

Which condition ensures that the total mechanical energy in an SHM system remains conserved during the motion?

**Forced oscillations** result when external periodic driving force F = F₀ cos(ω_d t) acts on oscillator, steady-state frequency equals driving frequency ω_d, amplitude A = F₀/√((k - m ω_d²)² + (b ω_d)²) depends on proximity to natural frequency ω₀ = √(k/m). Resonance when ω_d ≈ ω₀, amplitude maximum. Total mechanical energy (kinetic + potential) is conserved in SHM when no external dissipative forces (e.g., friction) act, allowing energy to transform without loss. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result Absence of dissipative forces follows, reflecting SHM dependence on amplitude A, ω and system parameters.

Ref: NCERT > Physics Book > Oscillations > Forced Oscillations and Resonance

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Forced Oscillations and Resonance

A particle in SHM has \( x = 3 \cos (2\pi t + \frac{\pi}{3}) \) (in m). What is its speed at \( t = 0.5 \, \text{s} \)?

**Resonance phenomenon** amplifies response when driving frequency matches natural frequency ω_d ≈ ω₀, large amplitude even with small F₀, as damping limits growth. Natural frequency determined by system parameters, resonance condition crucial for understanding vibrations and energy absorption, e.g., bridge collapse, tuning. Velocity: v = -ω A sin (ω t + Φ) . A = 3 m, ω = 2π s⁻¹, Φ = (π/3) . At t = 0.5 : 2π × 0.5 + (π/3) = π + (π/3) = (4π/3) . v = -2π × 3 sin (4π/3) = -6π sin (180° - 60°) = -6π (-(√(3)/2)) ≈ 16.31 m/s . Applying x =

Ref: NCERT > Physics Book > Oscillations > Forced Oscillations and Resonance

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Forced Oscillations and Resonance

A particle’s displacement is \( x = 5 \sin (2\pi t + \frac{\pi}{6}) \) (in m). What is its velocity at \( t = 0.5 \, \te

**Driven system** exhibits amplitude-frequency response peaking at resonance, phase shift between drive and displacement varying 0° to 180° across resonance. Forced oscillations sustain motion against damping, amplitude controlled by detuning |ω_d - ω₀| and damping strength b. Velocity: v = ω A cos (ω t + Φ) . A = 5 m, ω = 2π s⁻¹, Φ = (π/6) . At t = 0.5 : 2π × 0.5 + (π/6) = π + (π/6) = (7π/6) . v = 2π × 5 cos (7π/6) = 10π cos (180° - 30°) = 10π (-(√(3)/2)) ≈ -27.14 m/s . Applying x = A cos(ωt + φ), v

Ref: NCERT > Physics Book > Oscillations > Forced Oscillations and Resonance

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Forced Oscillations and Resonance

A particle in SHM has \( a = -64 x \) (in SI units). What is its period?

**Resonance phenomenon** amplifies response when driving frequency matches natural frequency ω_d ≈ ω₀, large amplitude even with small F₀, as damping limits growth. Natural frequency determined by system parameters, resonance condition crucial for understanding vibrations and energy absorption, e.g., bridge collapse, tuning. For SHM, a = -ω² x . Given a = -64 x , ω² = 64 ⇒ ω = 8 rad/s . Period: T = (2π/ω) = (2π/8) = (π/4) ≈ 0.785 s . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 0.785 s follows, reflecting SHM dependence

Ref: NCERT > Physics Book > Oscillations > Forced Oscillations and Resonance

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Damped Oscillations

A particle in SHM follows \( x = 2 \sin (4t + \frac{\pi}{2}) \) (in m). What is its acceleration at \( t = 0 \, \text{s}

**Real oscillators** experience damping, amplitude decreasing with time. Critical damping returns to equilibrium fastest without oscillation, overdamping slows return, underdamping shows decaying oscillations, classification based on b relative to 2mω₀, important for practical systems. Acceleration: a = -ω² x . ω = 4 s⁻¹, x(0) = 2 sin ((π/2)) = 2 × 1 = 2 m . a = -4² × 2 = -16 × 2 = -32 m/s² . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result -32 m/s² follows, reflecting SHM dependence on amplitude A, ω and system parameters.