A particle having mass 10 g oscillates according to the equation x = (2.0 cm) sin[(100 s-1)t + π/6]. Find (a) the amplitude, the time period and the spring constant (b) the position, the velocity and the acceleration at t = 0.

Sol. x = (2.0cm)sin [(100s–1) t + (π/6)] m = 10g. a) Amplitude = 2cm. ω = 100 sec–1 ∴ T = 2π/100 = π/50 sec = 0.063 sec. We know that T = 2π √(m/k) ⇒ T2 = 4π2 × m/k ⇒ k = 4 π2/T2 m = 105 dyne/cm = 100 N/m. [because ω = 2π/T = 100 sec-1] (b) At = 0 x = 2cm sin (π/6) = 2 × (1/2) = 1 cm. from the mean position. We know that x = A sin (ωt + ∅) v = A cos (ωt + ∅) = 2 × 100 cos (0 + π/6) = 200 × √3/2 = 100 √3 sec-1 = 1.73m/s c) a = – ω2 x = 1002 × 1 = 100 m/s2