To find the period of a particle in simple harmonic motion (SHM) with a maximum velocity of 0.5 m/s and an amplitude of 10 cm, the relevant equations include max velocity (v_max = ωA) and the period (T = 2π/ω). The relationship between maximum velocity and angular frequency (ω) allows for the calculation of ω as ω = v_max / A. Substituting the values gives ω = 0.5 m/s / 0.1 m = 5 rad/s. Finally, using T = 2π/ω results in a period of 1.26 seconds.