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