Forced Oscillations and Resonance

Forced Oscillations and Resonance

What is the significance of the phase constant in the displacement equation of SHM?

**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. The phase constant ( Φ in x = A cos (ω t + Φ) ) determines the initial position and velocity, setting the starting point of the oscillation cycle. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result It fixes the initial position

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Forced Oscillations and Resonance

A particle’s motion is \( x = 4 \sin (3t - \frac{\pi}{3}) \) (in m). What is its velocity at \( t = \frac{\pi}{6} \, \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 = 4 m, ω = 3 s⁻¹, Φ = -(π/3) . At t = (π/6) : 3 × (π/6) - (π/3) = (π/2) - (π/3) = (π/6) . v = 3 × 4 cos (π/6) = 12 × (√(3)/2) = 6√(3) ≈ 10.39 m/s . Applying x = A cos(ωt + φ), v = -ωA sin(ωt

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Forced Oscillations and Resonance

A particle in SHM has \( a = -36 x \) (in SI units). What is its frequency?

**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 = -36 x , ω² = 36 ⇒ ω = 6 rad/s . Frequency: v = (ω/2π) = (6/2 × 3.14) ≈ 0.955 Hz . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 0.955 Hz follows, reflecting SHM dependence

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Forced Oscillations and Resonance

Which of the following periodic motions cannot be classified as oscillatory due to the absence of a fixed equilibrium po

**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. Rotational motion of a ceiling fan is periodic but not oscillatory, as it lacks a fixed equilibrium point about which it moves to-and-fro, unlike SHM examples. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result Rotational motion of a ceiling fan follows, reflecting

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Forced Oscillations and Resonance

A pendulum has \( L = 1.4 \, \text{m}, g = 9.8 \, \text{m/s}^2 \). What is its angular frequency?

**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. ω = √((g/L)) = √((9.8/1.4)) ≈ √(7) ≈ 2.65 rad/s . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 2.65 rad/s follows, reflecting SHM dependence on amplitude A, ω and system parameters.

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Forced Oscillations and Resonance

What underlies the periodic nature of SHM when expressed as a superposition of sine and cosine functions?

**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. The periodicity arises from the repeating nature of sine and cosine functions, which have a fixed period ( 2π/ω ), ensuring the motion repeats consistently. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result The periodicity of trigonometric functions follows, reflecting SHM dependence on amplitude A, ω and system parameters.

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Forced Oscillations and Resonance

A pendulum of length \( 0.36 \, \text{m} \) oscillates with \( g = 9.8 \, \text{m/s}^2 \). What is its frequency?

**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. Period: T = 2π √((L/g)) = 2π √((0.36/9.8)) ≈ 2 × 3.14 √(0.0367) ≈ 1.2 s . Frequency: v = (1/T) = (1/1.2) ≈ 0.833 Hz . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 0.833 Hz follows, reflecting SHM dependence on

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Forced Oscillations and Resonance

A pendulum has \( L = 0.6 \, \text{m}, g = 9.8 \, \text{m/s}^2 \). What is its angular frequency?

**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. ω = √((g/L)) = √((9.8/0.6)) ≈ √(16.33) ≈ 4.04 rad/s . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 4.04 rad/s follows, reflecting SHM dependence on amplitude A, ω and system parameters.

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Forced Oscillations and Resonance

A particle in SHM has \( x = 4 \cos (2t + \frac{\pi}{3}) \) (in m). What is its speed at \( t = 0.5 \, \text{s} \)? (Tak

**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 = 4 m, ω = 2 s⁻¹, Φ = (π/3) . At t = 0.5 : 2 × 0.5 + (π/3) = 1 + (π/3) ≈ 2.047 rad ≈ 117° . v = -2 × 4 sin 117° ≈ -8 sin (180° - 63°) ≈ -8 × 0.838 ≈ -6.7 m/s

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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.

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Forced Oscillations and Resonance

Why does the period of a spring-mass system remain unaffected by changes in gravitational field strength?

**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. The period T = 2π √((m/k)) depends only on mass and spring constant, not gravity, which affects pendulums but not spring systems. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result The restoring force is independent of gravity follows, reflecting SHM dependence on amplitude A, ω and system parameters.

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Forced Oscillations and Resonance

A body oscillates with SHM according to \( x = 4 \cos (2\pi t + \frac{\pi}{6}) \) (in SI units). What is its velocity at

**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(t) = -ω A sin (ω t + Φ) . Here, A = 4 m, ω = 2π s⁻¹, Φ = (π/6) . At t = 0.5 s : ω t + Φ = 2π × 0.5 + (π/6) = π + (π/6) = (7π/6) . sin (7π/6) = sin (180° + 30°) = -sin 30° = -(1/2) . v = -2π

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