The motion of a particle executing simple harmonic motion is described by the displacement function, x(t) = A cos (ωt + φ). If the initial (t = 0) position of the particle is 1 cm and its initial velocity is ω cm/s, what are its amplitude and initial phase angle? The angular frequency of the particle is π s⁻¹. If instead of the cosine function, we choose the sine function to describe the SHM: x = B sin (ωt + α), what are the amplitude and initial phase of the particle with the above initial conditions.
Amplitude A = cm, Initial phase angle φ = -π/4 (or 7π/4 radians)
Amplitude B = cm, Initial phase angle α = π/4 radians
Explanation
This problem tests understanding of SHM equations and initial conditions. For the cosine function, we use the displacement x(t) = A cos(ωt + φ) and velocity v(t) = -ωA sin(ωt + φ). At t=0, we get two equations which we solve simultaneously to find A and φ. The same approach applies for the sine function, noting that velocity becomes v(t) = ωB cos(ωt + α). The key insight is that both representations describe the same physical motion, just with different phase constants.