#oscillation period

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 spring-mass system has \( m = 0.25 \, \text{kg} \) and \( k = 100 \, \text{N/m} \). What is its period of oscillation?

**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. Period: T = 2π √((m/k)) = 2π √((0.25/100)) = 2π √(0.0025) = 2π × 0.05 ≈ 0.314 s . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 0.314 s follows, reflecting SHM dependence on amplitude A, ω and system parameters.

Ref: NCERT > Physics Book > Oscillations > Damped Oscillations

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Equations of SHM, Phase and Angular Frequency

Two identical springs (\( k = 80 \, \text{N/m} \)) are attached to a \( 2 \, \text{kg} \) mass as in Fig. 13.14. What is

**SHM representation** using sine or cosine equivalent with phase offset, ω relates to system parameters like mass and stiffness. Understanding ω and φ permits prediction of position at any time and comparison of two SHM via phase difference Δφ = φ₂ - φ₁. Effective kₑff = 2k = 2 × 80 = 160 N/m . T = 2π √((m/kₑff)) = 2π √((2/160)) = 2π √(0.0125) ≈ 0.702 s . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 0.702 s follows, reflecting SHM dependence on amplitude A, ω and system parameters.

Ref: NCERT > Physics Book > Oscillations > Equations of SHM, Phase and Angular Frequency

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Spring-Mass System and Combination of Springs

A spring-mass system oscillates with \( T = 0.7 \, \text{s} \) when \( m = 0.7 \, \text{kg} \). What is the spring const

**Oscillation of mass attached to spring** follows Hooke's law F = -k x, SHM with ω = √(k/m). Effective stiffness for two identical springs in parallel doubles, k_eff = 2k, increasing frequency by √2, while series halves stiffness to k/2, lowering frequency. Energy E = ½ k_eff A². T = 2π √((m/k)) . 0.7 = 2π √((0.7/k)) ⇒ (0.7/2π) = √((0.7/k)) . (0.1114)² = (0.7/k) ⇒ k = (0.7/0.01241) ≈ 56.4 N/m . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 56.4 N/m follows, reflecting SHM dependence on amplitude A, ω and system parameters.