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#position function

3 public questions tagged with this topic.

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

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

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Damped Oscillations

A particle’s x-projection from circular motion is \( x = 6 \cos (3t) \) (in m). What is its maximum acceleration?

**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. Maximum acceleration: aₘₐₓ = ω² A . A = 6 m, ω = 3 s⁻¹ . aₘₐₓ = 3² × 6 = 9 × 6 = 54 m/s² . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 54 m/s² 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

A particle in SHM has \( x = 7 \sin (3\pi t + \frac{\pi}{3}) \) (in m). What is its speed at \( t = 0.5 \, \text{s} \)?

**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. Velocity: v = ω A cos (ω t + Φ) . A = 7 m, ω = 3π s⁻¹, Φ = (π/3) . At t = 0.5 : 3π × 0.5 + (π/3) = (3π/2) + (π/3) = (9π/6) + (2π/6) = (11π/6) . v = 3π × 7 cos (11π/6) = 21π cos (180° - 30°) = 21π × (√(3)/2) ≈

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

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