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#displacement equation

3 public questions tagged with this topic.

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

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

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Equations of SHM, Phase and Angular Frequency

A particle’s displacement is \( x = 5 \sin (\pi t + \frac{\pi}{3}) \) (in m). What is its velocity at \( t = 0 \, \text{

**Angular frequency** ω = 2πf = 2π/T characterizes rapidity, independent of amplitude for SHM. Phase constant φ shifts sine/cosine, allowing any initial condition, e.g., x(0)=0 requires φ=0 for sine form. Displacement, velocity, acceleration share ω but differ in phase by 90° and 180°. Velocity: v = ω A cos (ω t + Φ) . A = 5 m, ω = π s⁻¹, Φ = (π/3) . At t = 0 : v = π × 5 cos ((π/3)) = 5π × 0.5 = 2.5π ≈ 7.85 m/s . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and

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

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Displacement, Velocity and Acceleration in SHM

A particle in SHM has \( x = 2 \sin (4t + \frac{\pi}{3}) \) (in m). What is its acceleration at \( t = 0 \, \text{s} \)?

**SHM kinematics** given by x = A cos(ωt + φ), velocity v = dx/dt = -ω A sin(ωt + φ), acceleration a = dv/dt = -ω² A cos(ωt + φ) = -ω² x, maxima v_max = ωA at mean position x=0, a_max = ω²A at extremes x=±A. Phase φ determines initial position, ω = 2π/T = √(k/m). Acceleration: a = -ω² x . ω = 4 s⁻¹, x(0) = 2 sin (π/3) = 2 × (√(3)/2) = √(3) ≈ 1.732 m . a = -4² × 1.732 = -16 × 1.732 ≈ -27.71 m/s² . Applying x = A cos(ωt + φ), v = -ωA

Ref: NCERT > Physics Book > Oscillations > Displacement, Velocity and Acceleration in SHM

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