Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial (t = 0) position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: (x is in cm and t is in s).
x = –2 sin (3t + π/3)
Amplitude (radius) A = 2 cm, Angular speed ω = 3 rad/s, Initial phase φ = 5π/6. At t=0, particle is at angle 5π/6 from positive x-axis (150° anticlockwise).
x = cos (π/6 – t)
Amplitude (radius) A = 1 cm, Angular speed ω = 1 rad/s, Initial phase φ = -π/6. At t=0, particle is at angle -π/6 (or 11π/6) from positive x-axis.
x = 3 sin (2π t + π/4)
Amplitude (radius) A = 3 cm, Angular speed ω = 2π rad/s, Initial phase φ = -π/4. At t=0, particle is at angle -π/4 (or 7π/4) from positive x-axis.
x = 2 cos π t
Amplitude (radius) A = 2 cm, Angular speed ω = π rad/s, Initial phase φ = 0. At t=0, particle is on positive x-axis.
Explanation
For each SHM equation, we convert to standard form x = A cos(ωt + φ). The reference circle has radius equal to amplitude A. The particle rotates anticlockwise with angular speed ω. At t=0, the position vector makes angle φ with positive x-axis. For (a): -sin(θ) = cos(θ + π/2), giving φ = 5π/6. For (b): cos(π/6 - t) = cos(t - π/6), giving φ = -π/6. For (c): sin(θ) = cos(θ - π/2), giving φ = -π/4. For (d): already in standard form with φ = 0.